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| author | julie <julielangou@users.noreply.github.com> | 2011-04-07 13:53:54 +0000 |
|---|---|---|
| committer | julie <julielangou@users.noreply.github.com> | 2011-04-07 13:53:54 +0000 |
| commit | 16973f0c15a4327f8a78b9652f178246895778c1 (patch) | |
| tree | 1d022faeffac8e1524f4f73517483f4cd828038d /SRC | |
| parent | 8b5dd5433b205f27821b10d5389f55bcd4a1e87a (diff) | |
| download | lapack-16973f0c15a4327f8a78b9652f178246895778c1.tar.gz lapack-16973f0c15a4327f8a78b9652f178246895778c1.tar.bz2 lapack-16973f0c15a4327f8a78b9652f178246895778c1.zip | |
Second and hopefully last pass to homgenize notation for transpose (**T) and conjugate transpose (**H)
Corresponds to bug0024
Please take a look and let me know if you find some old notation of transpose.
I am going to close bug0024.
Julie
Diffstat (limited to 'SRC')
262 files changed, 679 insertions, 678 deletions
diff --git a/SRC/cgebd2.f b/SRC/cgebd2.f index 6d85c003..13d56b58 100644 --- a/SRC/cgebd2.f +++ b/SRC/cgebd2.f @@ -17,7 +17,7 @@ * ======= * * CGEBD2 reduces a complex general m by n matrix A to upper or lower -* real bidiagonal form B by a unitary transformation: Q' * A * P = B. +* real bidiagonal form B by a unitary transformation: Q**H * A * P = B. * * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. * diff --git a/SRC/cgels.f b/SRC/cgels.f index 3fd0e065..ebd38463 100644 --- a/SRC/cgels.f +++ b/SRC/cgels.f @@ -299,9 +299,9 @@ * ELSE * -* Overdetermined system of equations A' * X = B +* Overdetermined system of equations A**H * X = B * -* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) +* B(1:N,1:NRHS) := inv(R**H) * B(1:N,1:NRHS) * CALL CTRTRS( 'Upper', 'Conjugate transpose','Non-unit', $ N, NRHS, A, LDA, B, LDB, INFO ) @@ -372,7 +372,7 @@ * ELSE * -* overdetermined system min || A' * X - B || +* overdetermined system min || A**H * X - B || * * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) * @@ -382,7 +382,7 @@ * * workspace at least NRHS, optimally NRHS*NB * -* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) +* B(1:M,1:NRHS) := inv(L**H) * B(1:M,1:NRHS) * CALL CTRTRS( 'Lower', 'Conjugate transpose', 'Non-unit', $ M, NRHS, A, LDA, B, LDB, INFO ) diff --git a/SRC/cgelsy.f b/SRC/cgelsy.f index 636b1709..e6052f66 100644 --- a/SRC/cgelsy.f +++ b/SRC/cgelsy.f @@ -43,8 +43,8 @@ * A * P = Q * [ T11 0 ] * Z * [ 0 0 ] * The minimum-norm solution is then -* X = P * Z' [ inv(T11)*Q1'*B ] -* [ 0 ] +* X = P * Z**H [ inv(T11)*Q1**H*B ] +* [ 0 ] * where Q1 consists of the first RANK columns of Q. * * This routine is basically identical to the original xGELSX except diff --git a/SRC/cggsvd.f b/SRC/cggsvd.f index e8306976..944b9e2b 100644 --- a/SRC/cggsvd.f +++ b/SRC/cggsvd.f @@ -210,7 +210,7 @@ * TOLA REAL * TOLB REAL * TOLA and TOLB are the thresholds to determine the effective -* rank of (A',B')**H. Generally, they are set to +* rank of (A**H,B**H)**H. Generally, they are set to * TOLA = MAX(M,N)*norm(A)*MACHEPS, * TOLB = MAX(P,N)*norm(B)*MACHEPS. * The size of TOLA and TOLB may affect the size of backward diff --git a/SRC/cherfs.f b/SRC/cherfs.f index 413fd7f3..dc1b8928 100644 --- a/SRC/cherfs.f +++ b/SRC/cherfs.f @@ -308,7 +308,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/chetd2.f b/SRC/chetd2.f index 541ac522..ad123ef4 100644 --- a/SRC/chetd2.f +++ b/SRC/chetd2.f @@ -237,7 +237,7 @@ CALL CAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**H +* A := A - v * w**H - w * v**H * CALL CHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, $ A( I+1, I+1 ), LDA ) diff --git a/SRC/chetrd.f b/SRC/chetrd.f index 2868958c..cf928151 100644 --- a/SRC/chetrd.f +++ b/SRC/chetrd.f @@ -237,7 +237,7 @@ $ LDWORK ) * * Update the unreduced submatrix A(1:i-1,1:i-1), using an -* update of the form: A := A - V*W' - W*V**H +* update of the form: A := A - V*W**H - W*V**H * CALL CHER2K( UPLO, 'No transpose', I-1, NB, -CONE, $ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA ) @@ -268,7 +268,7 @@ $ TAU( I ), WORK, LDWORK ) * * Update the unreduced submatrix A(i+nb:n,i+nb:n), using -* an update of the form: A := A - V*W' - W*V**H +* an update of the form: A := A - V*W**H - W*V**H * CALL CHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE, $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, diff --git a/SRC/chetri2x.f b/SRC/chetri2x.f index 9bafd618..aaa6a85e 100644 --- a/SRC/chetri2x.f +++ b/SRC/chetri2x.f @@ -156,7 +156,7 @@ IF( UPPER ) THEN * -* invA = P * inv(U**H)*inv(D)*inv(U)*P'. +* invA = P * inv(U**H)*inv(D)*inv(U)*P**H. * CALL CTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -269,7 +269,7 @@ END IF END DO * -* U11T*invD1*U11->U11 +* U11**H*invD1*U11->U11 * CALL CTRMM('L','U','C','U',NNB, NNB, $ CONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -280,12 +280,12 @@ END DO END DO * -* U01'invD*U01->A(CUT+I,CUT+J) +* U01**H*invD*U01->A(CUT+I,CUT+J) * CALL CGEMM('C','N',NNB,NNB,CUT,CONE,A(1,CUT+1),LDA, $ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* U11 = U11T*invD1*U11 + U01'invD*U01 +* U11 = U11**H*invD1*U11 + U01**H*invD*U01 * DO I=1,NNB DO J=I,NNB @@ -293,7 +293,7 @@ END DO END DO * -* U01 = U00T*invD0*U01 +* U01 = U00**H*invD0*U01 * CALL CTRMM('L',UPLO,'C','U',CUT, NNB, $ CONE,A,LDA,WORK,N+NB+1) @@ -311,7 +311,7 @@ * END DO * -* Apply PERMUTATIONS P and P': P * inv(U**H)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**H: P * inv(U**H)*inv(D)*inv(U) *P**H * I=1 DO WHILE ( I .LE. N ) @@ -333,7 +333,7 @@ * * LOWER... * -* invA = P * inv(U**H)*inv(D)*inv(U)*P'. +* invA = P * inv(U**H)*inv(D)*inv(U)*P**H. * CALL CTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -440,7 +440,7 @@ END IF END DO * -* L11T*invD1*L11->L11 +* L11**H*invD1*L11->L11 * CALL CTRMM('L',UPLO,'C','U',NNB, NNB, $ CONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -453,13 +453,13 @@ * IF ( (CUT+NNB) .LT. N ) THEN * -* L21T*invD2*L21->A(CUT+I,CUT+J) +* L21**H*invD2*L21->A(CUT+I,CUT+J) * CALL CGEMM('C','N',NNB,NNB,N-NNB-CUT,CONE,A(CUT+NNB+1,CUT+1) $ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* L11 = L11T*invD1*L11 + U01'invD*U01 +* L11 = L11**H*invD1*L11 + U01**H*invD*U01 * DO I=1,NNB DO J=1,I @@ -467,7 +467,7 @@ END DO END DO * -* L01 = L22T*invD2*L21 +* L01 = L22**H*invD2*L21 * CALL CTRMM('L',UPLO,'C','U', N-NNB-CUT, NNB, $ CONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1) @@ -480,7 +480,7 @@ END DO ELSE * -* L11 = L11T*invD1*L11 +* L11 = L11**H*invD1*L11 * DO I=1,NNB DO J=1,I @@ -494,7 +494,7 @@ CUT=CUT+NNB END DO * -* Apply PERMUTATIONS P and P': P * inv(U**H)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**H: P * inv(U**H)*inv(D)*inv(U) *P**H * I=N DO WHILE ( I .GE. 1 ) diff --git a/SRC/chfrk.f b/SRC/chfrk.f index 8e85bfc5..07b909a9 100644 --- a/SRC/chfrk.f +++ b/SRC/chfrk.f @@ -26,11 +26,11 @@ * * CHFRK performs one of the Hermitian rank--k operations * -* C := alpha*A*conjg( A' ) + beta*C, +* C := alpha*A*A**H + beta*C, * * or * -* C := alpha*conjg( A' )*A + beta*C, +* C := alpha*A**H*A + beta*C, * * where alpha and beta are real scalars, C is an n--by--n Hermitian * matrix and A is an n--by--k matrix in the first case and a k--by--n @@ -60,9 +60,9 @@ * On entry, TRANS specifies the operation to be performed as * follows: * -* TRANS = 'N' or 'n' C := alpha*A*conjg( A' ) + beta*C. +* TRANS = 'N' or 'n' C := alpha*A*A**H + beta*C. * -* TRANS = 'C' or 'c' C := alpha*conjg( A' )*A + beta*C. +* TRANS = 'C' or 'c' C := alpha*A**H*A + beta*C. * * Unchanged on exit. * diff --git a/SRC/chprfs.f b/SRC/chprfs.f index 5bcff58f..a84b3a8e 100644 --- a/SRC/chprfs.f +++ b/SRC/chprfs.f @@ -306,7 +306,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL CHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/chptrd.f b/SRC/chptrd.f index 82d2fd2f..94357958 100644 --- a/SRC/chptrd.f +++ b/SRC/chptrd.f @@ -217,7 +217,7 @@ CALL CAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**H +* A := A - v * w**H - w * v**H * CALL CHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, $ AP( I1I1 ) ) diff --git a/SRC/cla_gbrcond_c.f b/SRC/cla_gbrcond_c.f index 209111bf..803a8544 100644 --- a/SRC/cla_gbrcond_c.f +++ b/SRC/cla_gbrcond_c.f @@ -218,7 +218,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/cla_gbrcond_x.f b/SRC/cla_gbrcond_x.f index bea99e6d..54746ea4 100644 --- a/SRC/cla_gbrcond_x.f +++ b/SRC/cla_gbrcond_x.f @@ -201,7 +201,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/cla_gercond_c.f b/SRC/cla_gercond_c.f index 92403302..913172a3 100644 --- a/SRC/cla_gercond_c.f +++ b/SRC/cla_gercond_c.f @@ -194,7 +194,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/cla_gercond_x.f b/SRC/cla_gercond_x.f index 405f5e15..b50faa89 100644 --- a/SRC/cla_gercond_x.f +++ b/SRC/cla_gercond_x.f @@ -177,7 +177,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/cla_hercond_c.f b/SRC/cla_hercond_c.f index 5e9d850c..3e5506b0 100644 --- a/SRC/cla_hercond_c.f +++ b/SRC/cla_hercond_c.f @@ -203,7 +203,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/cla_hercond_x.f b/SRC/cla_hercond_x.f index d5df6542..85880d7d 100644 --- a/SRC/cla_hercond_x.f +++ b/SRC/cla_hercond_x.f @@ -179,7 +179,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/cla_porcond_c.f b/SRC/cla_porcond_c.f index 492eaa9e..b6f2fc68 100644 --- a/SRC/cla_porcond_c.f +++ b/SRC/cla_porcond_c.f @@ -199,7 +199,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/cla_porcond_x.f b/SRC/cla_porcond_x.f index 9e7d945e..2ed1a6ff 100644 --- a/SRC/cla_porcond_x.f +++ b/SRC/cla_porcond_x.f @@ -174,7 +174,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/cla_syrcond_c.f b/SRC/cla_syrcond_c.f index 8e1864d7..291ade08 100644 --- a/SRC/cla_syrcond_c.f +++ b/SRC/cla_syrcond_c.f @@ -204,7 +204,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/cla_syrcond_x.f b/SRC/cla_syrcond_x.f index b07a3077..eee2ac1b 100644 --- a/SRC/cla_syrcond_x.f +++ b/SRC/cla_syrcond_x.f @@ -180,7 +180,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**T). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/clacn2.f b/SRC/clacn2.f index f01dda06..5b76cde4 100644 --- a/SRC/clacn2.f +++ b/SRC/clacn2.f @@ -33,8 +33,8 @@ * X (input/output) COMPLEX array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, -* where A' is the conjugate transpose of A, and CLACN2 must be +* A**H * X, if KASE=2, +* where A**H is the conjugate transpose of A, and CLACN2 must be * re-called with all the other parameters unchanged. * * EST (input/output) REAL @@ -45,7 +45,7 @@ * KASE (input/output) INTEGER * On the initial call to CLACN2, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**H * X. * On the final return from CLACN2, KASE will again be 0. * * ISAVE (input/output) INTEGER array, dimension (3) diff --git a/SRC/clacon.f b/SRC/clacon.f index 0c4ee14e..e0bb02f5 100644 --- a/SRC/clacon.f +++ b/SRC/clacon.f @@ -32,8 +32,8 @@ * X (input/output) COMPLEX array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, -* where A' is the conjugate transpose of A, and CLACON must be +* A**H * X, if KASE=2, +* where A**H is the conjugate transpose of A, and CLACON must be * re-called with all the other parameters unchanged. * * EST (input/output) REAL @@ -44,7 +44,7 @@ * KASE (input/output) INTEGER * On the initial call to CLACON, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**H * X. * On the final return from CLACON, KASE will again be 0. * * Further Details diff --git a/SRC/claesy.f b/SRC/claesy.f index c98058bb..6aaf5149 100644 --- a/SRC/claesy.f +++ b/SRC/claesy.f @@ -127,7 +127,7 @@ * * Choose CS1 = 1 and SN1 to satisfy the first equation, then * scale the components of this eigenvector so that the matrix -* of eigenvectors X satisfies X * X' = I . (No scaling is +* of eigenvectors X satisfies X * X**T = I . (No scaling is * done if the norm of the eigenvalue matrix is less than THRESH.) * SN1 = ( RT1-A ) / B diff --git a/SRC/clahr2.f b/SRC/clahr2.f index 9c002c2b..1fd6af28 100644 --- a/SRC/clahr2.f +++ b/SRC/clahr2.f @@ -151,7 +151,7 @@ $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 ) CALL CLACGV( I-1, A( K+I-1, 1 ), LDA ) * -* Apply I - V * T' * V**H to this column (call it b) from the +* Apply I - V * T**H * V**H to this column (call it b) from the * left, using the last column of T as workspace * * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) @@ -159,7 +159,7 @@ * * where V1 is unit lower triangular * -* w := V1' * b1 +* w := V1**H * b1 * CALL CCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL CTRMV( 'Lower', 'Conjugate transpose', 'UNIT', diff --git a/SRC/claic1.f b/SRC/claic1.f index f17c2b61..32bf0212 100644 --- a/SRC/claic1.f +++ b/SRC/claic1.f @@ -28,15 +28,15 @@ * [ s*x ] * xhat = [ c ] * is an approximate singular vector of -* [ L 0 ] -* Lhat = [ w' gamma ] +* [ L 0 ] +* Lhat = [ w**H gamma ] * in the sense that * twonorm(Lhat*xhat) = sestpr. * * Depending on JOB, an estimate for the largest or smallest singular * value is computed. * -* Note that [s c]' and sestpr**2 is an eigenpair of the system +* Note that [s c]**H and sestpr**2 is an eigenpair of the system * * diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ] * [ conjg(gamma) ] diff --git a/SRC/clalsa.f b/SRC/clalsa.f index 978892e3..cac8ff97 100644 --- a/SRC/clalsa.f +++ b/SRC/clalsa.f @@ -79,7 +79,7 @@ * POLES, GIVNUM, and Z. * * VT (input) REAL array, dimension ( LDU, SMLSIZ+1 ). -* On entry, VT' contains the right singular vector matrices of +* On entry, VT**H contains the right singular vector matrices of * all subproblems at the bottom level. * * K (input) INTEGER array, dimension ( N ). diff --git a/SRC/clalsd.f b/SRC/clalsd.f index da84ca19..98812577 100644 --- a/SRC/clalsd.f +++ b/SRC/clalsd.f @@ -287,7 +287,7 @@ * * Since B is complex, the following call to SGEMM is performed * in two steps (real and imaginary parts). That is for V * B -* (in the real version of the code V' is stored in WORK). +* (in the real version of the code V**H is stored in WORK). * * CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, * $ WORK( NWORK ), N ) diff --git a/SRC/clanhf.f b/SRC/clanhf.f index fb89fd5a..621a3028 100644 --- a/SRC/clanhf.f +++ b/SRC/clanhf.f @@ -1026,7 +1026,7 @@ ELSE * A is xpose & A is k by n IF( ILU.EQ.0 ) THEN -* A' is upper +* A**H is upper DO J = 1, K - 2 CALL CLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE, S ) * U at A(0,k) @@ -1080,7 +1080,7 @@ L = L + LDA + 1 END DO ELSE -* A' is lower +* A**H is lower DO J = 1, K - 1 CALL CLASSQ( J, A( 0+J*LDA ), 1, SCALE, S ) * U at A(0,0) @@ -1215,7 +1215,7 @@ ELSE * A is xpose IF( ILU.EQ.0 ) THEN -* A' is upper +* A**H is upper DO J = 1, K - 1 CALL CLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE, S ) * U at A(0,k+1) @@ -1281,7 +1281,7 @@ END IF END IF ELSE -* A' is lower +* A**H is lower DO J = 1, K - 1 CALL CLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE, S ) * U at A(0,1) diff --git a/SRC/claqr5.f b/SRC/claqr5.f index 9f4bc77c..5ea34429 100644 --- a/SRC/claqr5.f +++ b/SRC/claqr5.f @@ -634,14 +634,14 @@ CALL CLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ), $ LDH, WH( KZS+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**H ==== * CALL CLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH ) CALL CTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE, $ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ), $ LDWH ) * -* ==== Multiply top of H by U11' ==== +* ==== Multiply top of H by U11**H ==== * CALL CGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU, $ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH ) @@ -651,7 +651,7 @@ CALL CLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH, $ WH( I2+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**H ==== * CALL CTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE, $ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH ) diff --git a/SRC/clarfb.f b/SRC/clarfb.f index c4980bd0..c9771be7 100644 --- a/SRC/clarfb.f +++ b/SRC/clarfb.f @@ -165,7 +165,7 @@ * * W := C**H * V = (C1**H * V1 + C2**H * V2) (stored in WORK) * -* W := C1' +* W := C1**H * DO 10 J = 1, K CALL CCOPY( LASTC, C( J, 1 ), LDC, WORK( 1, J ), 1 ) diff --git a/SRC/clarzt.f b/SRC/clarzt.f index 3abd29b0..1fc319a1 100644 --- a/SRC/clarzt.f +++ b/SRC/clarzt.f @@ -27,12 +27,12 @@ * If STOREV = 'C', the vector which defines the elementary reflector * H(i) is stored in the i-th column of the array V, and * -* H = I - V * T * V' +* H = I - V * T * V**H * * If STOREV = 'R', the vector which defines the elementary reflector * H(i) is stored in the i-th row of the array V, and * -* H = I - V' * T * V +* H = I - V**H * T * V * * Currently, only STOREV = 'R' and DIRECT = 'B' are supported. * diff --git a/SRC/clasr.f b/SRC/clasr.f index f03cb656..5f18a4b2 100644 --- a/SRC/clasr.f +++ b/SRC/clasr.f @@ -274,7 +274,7 @@ END IF ELSE IF( LSAME( SIDE, 'R' ) ) THEN * -* Form A * P' +* Form A * P**T * IF( LSAME( PIVOT, 'V' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN diff --git a/SRC/clatbs.f b/SRC/clatbs.f index 63036dad..7af3c258 100644 --- a/SRC/clatbs.f +++ b/SRC/clatbs.f @@ -24,7 +24,7 @@ * A * x = s*b, A**T * x = s*b, or A**H * x = s*b, * * with scaling to prevent overflow, where A is an upper or lower -* triangular band matrix. Here A' denotes the transpose of A, x and b +* triangular band matrix. Here A**T denotes the transpose of A, x and b * are n-element vectors, and s is a scaling factor, usually less than * or equal to 1, chosen so that the components of x will be less than * the overflow threshold. If the unscaled problem will not cause diff --git a/SRC/cpbrfs.f b/SRC/cpbrfs.f index 49b7ef4c..7c4d0fcf 100644 --- a/SRC/cpbrfs.f +++ b/SRC/cpbrfs.f @@ -311,7 +311,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL CPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/cpbstf.f b/SRC/cpbstf.f index 2055c658..fe1cd307 100644 --- a/SRC/cpbstf.f +++ b/SRC/cpbstf.f @@ -83,19 +83,19 @@ * * on entry: on exit: * -* * * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75' -* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76' -* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 +* * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H +* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H +* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 * * If UPLO = 'L', the array AB holds: * * on entry: on exit: * -* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 -* a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * -* a31 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * * +* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 +* a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 * +* a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * * * -* Array elements marked * are not used by the routine; s12' denotes +* Array elements marked * are not used by the routine; s12**H denotes * conjg(s12); the diagonal elements of S are real. * * ===================================================================== diff --git a/SRC/cporfs.f b/SRC/cporfs.f index 9643bd91..b8b1ae2a 100644 --- a/SRC/cporfs.f +++ b/SRC/cporfs.f @@ -302,7 +302,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL CPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/cpprfs.f b/SRC/cpprfs.f index 0e511914..871d9717 100644 --- a/SRC/cpprfs.f +++ b/SRC/cpprfs.f @@ -300,7 +300,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL CPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/cppsvx.f b/SRC/cppsvx.f index 0997cdef..8165703d 100644 --- a/SRC/cppsvx.f +++ b/SRC/cppsvx.f @@ -46,7 +46,7 @@ * A = U**H * U , if UPLO = 'U', or * A = L * L**H, if UPLO = 'L', * where U is an upper triangular matrix, L is a lower triangular -* matrix, and ' indicates conjugate transpose. +* matrix, and **H indicates conjugate transpose. * * 3. If the leading i-by-i principal minor is not positive definite, * then the routine returns with INFO = i. Otherwise, the factored diff --git a/SRC/cpstf2.f b/SRC/cpstf2.f index 3369ac87..4684c6c0 100644 --- a/SRC/cpstf2.f +++ b/SRC/cpstf2.f @@ -22,8 +22,8 @@ * pivoting of a complex Hermitian positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**H * U , if UPLO = 'U', +* P**T * A * P = L * L**H, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -162,7 +162,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**H * U * DO 150 J = 1, N * @@ -234,7 +234,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**H * DO 180 J = 1, N * diff --git a/SRC/cpstrf.f b/SRC/cpstrf.f index 9c958c43..d998d109 100644 --- a/SRC/cpstrf.f +++ b/SRC/cpstrf.f @@ -22,8 +22,8 @@ * pivoting of a complex Hermitian positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**H * U , if UPLO = 'U', +* P**T * A * P = L * L**H, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -172,7 +172,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**H * U * DO 160 K = 1, N, NB * @@ -267,7 +267,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**H * DO 210 K = 1, N, NB * diff --git a/SRC/cptcon.f b/SRC/cptcon.f index e60b41b2..c5da88c0 100644 --- a/SRC/cptcon.f +++ b/SRC/cptcon.f @@ -118,7 +118,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**H. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H. * * Solve M(L) * x = e. * diff --git a/SRC/cptrfs.f b/SRC/cptrfs.f index 88e64504..58c08e09 100644 --- a/SRC/cptrfs.f +++ b/SRC/cptrfs.f @@ -327,7 +327,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**H. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H. * * Solve M(L) * x = e. * @@ -19,7 +19,7 @@ * * CSPR performs the symmetric rank 1 operation * -* A := alpha*x*conjg( x' ) + A, +* A := alpha*x*x**H + A, * * where alpha is a complex scalar, x is an n element vector and A is an * n by n symmetric matrix, supplied in packed form. diff --git a/SRC/csprfs.f b/SRC/csprfs.f index 71a27132..3eae840d 100644 --- a/SRC/csprfs.f +++ b/SRC/csprfs.f @@ -305,7 +305,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL CSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) DO 110 I = 1, N @@ -19,7 +19,7 @@ * * CSYR performs the symmetric rank 1 operation * -* A := alpha*x*( x' ) + A, +* A := alpha*x*x**H + A, * * where alpha is a complex scalar, x is an n element vector and A is an * n by n symmetric matrix. diff --git a/SRC/csyrfs.f b/SRC/csyrfs.f index beeb98b1..c7c203c9 100644 --- a/SRC/csyrfs.f +++ b/SRC/csyrfs.f @@ -308,7 +308,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL CSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/csytri2x.f b/SRC/csytri2x.f index 489bf452..db02d01a 100644 --- a/SRC/csytri2x.f +++ b/SRC/csytri2x.f @@ -154,7 +154,7 @@ IF( UPPER ) THEN * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL CTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -267,7 +267,7 @@ END IF END DO * -* U11T*invD1*U11->U11 +* U11**T*invD1*U11->U11 * CALL CTRMM('L','U','T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -278,12 +278,12 @@ END DO END DO * -* U01'invD*U01->A(CUT+I,CUT+J) +* U01**T*invD*U01->A(CUT+I,CUT+J) * CALL CGEMM('T','N',NNB,NNB,CUT,ONE,A(1,CUT+1),LDA, $ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* U11 = U11T*invD1*U11 + U01'invD*U01 +* U11 = U11**T*invD1*U11 + U01**T*invD*U01 * DO I=1,NNB DO J=I,NNB @@ -291,7 +291,7 @@ END DO END DO * -* U01 = U00T*invD0*U01 +* U01 = U00**T*invD0*U01 * CALL CTRMM('L',UPLO,'T','U',CUT, NNB, $ ONE,A,LDA,WORK,N+NB+1) @@ -309,7 +309,7 @@ * END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=1 DO WHILE ( I .LE. N ) @@ -331,7 +331,7 @@ * * LOWER... * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL CTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -438,7 +438,7 @@ END IF END DO * -* L11T*invD1*L11->L11 +* L11**T*invD1*L11->L11 * CALL CTRMM('L',UPLO,'T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -451,13 +451,13 @@ * IF ( (CUT+NNB) .LT. N ) THEN * -* L21T*invD2*L21->A(CUT+I,CUT+J) +* L21**T*invD2*L21->A(CUT+I,CUT+J) * CALL CGEMM('T','N',NNB,NNB,N-NNB-CUT,ONE,A(CUT+NNB+1,CUT+1) $ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* L11 = L11T*invD1*L11 + U01'invD*U01 +* L11 = L11**T*invD1*L11 + U01**T*invD*U01 * DO I=1,NNB DO J=1,I @@ -465,7 +465,7 @@ END DO END DO * -* L01 = L22T*invD2*L21 +* L01 = L22**T*invD2*L21 * CALL CTRMM('L',UPLO,'T','U', N-NNB-CUT, NNB, $ ONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1) @@ -478,7 +478,7 @@ END DO ELSE * -* L11 = L11T*invD1*L11 +* L11 = L11**T*invD1*L11 * DO I=1,NNB DO J=1,I @@ -492,7 +492,7 @@ CUT=CUT+NNB END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=N DO WHILE ( I .GE. 1 ) diff --git a/SRC/ctbcon.f b/SRC/ctbcon.f index 302cc490..4d7e620e 100644 --- a/SRC/ctbcon.f +++ b/SRC/ctbcon.f @@ -148,7 +148,7 @@ RCOND = ZERO SMLNUM = SLAMCH( 'Safe minimum' )*REAL( MAX( N, 1 ) ) * -* Compute the 1-norm of the triangular matrix A or A'. +* Compute the 1-norm of the triangular matrix A or A**H. * ANORM = CLANTB( NORM, UPLO, DIAG, N, KD, AB, LDAB, RWORK ) * @@ -177,7 +177,7 @@ $ AB, LDAB, WORK, SCALE, RWORK, INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**H). * CALL CLATBS( UPLO, 'Conjugate transpose', DIAG, NORMIN, $ N, KD, AB, LDAB, WORK, SCALE, RWORK, INFO ) diff --git a/SRC/ctfsm.f b/SRC/ctfsm.f index 9934b6c6..841e1173 100644 --- a/SRC/ctfsm.f +++ b/SRC/ctfsm.f @@ -31,7 +31,7 @@ * where alpha is a scalar, X and B are m by n matrices, A is a unit, or * non-unit, upper or lower triangular matrix and op( A ) is one of * -* op( A ) = A or op( A ) = conjg( A' ). +* op( A ) = A or op( A ) = A**H. * * A is in Rectangular Full Packed (RFP) Format. * diff --git a/SRC/ctgexc.f b/SRC/ctgexc.f index 06f3c7a5..e0083b23 100644 --- a/SRC/ctgexc.f +++ b/SRC/ctgexc.f @@ -20,7 +20,7 @@ * * CTGEXC reorders the generalized Schur decomposition of a complex * matrix pair (A,B), using an unitary equivalence transformation -* (A, B) := Q * (A, B) * Z', so that the diagonal block of (A, B) with +* (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with * row index IFST is moved to row ILST. * * (A, B) must be in generalized Schur canonical form, that is, A and diff --git a/SRC/ctgsen.f b/SRC/ctgsen.f index de9b77e6..a9be1e9b 100644 --- a/SRC/ctgsen.f +++ b/SRC/ctgsen.f @@ -231,11 +231,11 @@ * where sigma-min(Zu) is the smallest singular value of the * (2*n1*n2)-by-(2*n1*n2) matrix * -* Zu = [ kron(In2, A11) -kron(A22', In1) ] -* [ kron(In2, B11) -kron(B22', In1) ]. +* Zu = [ kron(In2, A11) -kron(A22**H, In1) ] +* [ kron(In2, B11) -kron(B22**H, In1) ]. * -* Here, Inx is the identity matrix of size nx and A22' is the -* transpose of A22. kron(X, Y) is the Kronecker product between +* Here, Inx is the identity matrix of size nx and A22**H is the +* conjuguate transpose of A22. kron(X, Y) is the Kronecker product between * the matrices X and Y. * * When DIF(2) is small, small changes in (A, B) can cause large changes diff --git a/SRC/ctgsy2.f b/SRC/ctgsy2.f index a9fdf836..d03a5fc7 100644 --- a/SRC/ctgsy2.f +++ b/SRC/ctgsy2.f @@ -36,17 +36,17 @@ * In matrix notation solving equation (1) corresponds to solve * Zx = scale * b, where Z is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ], +* Z = [ kron(In, A) -kron(B**H, Im) ] (2) +* [ kron(In, D) -kron(E**H, Im) ], * -* Ik is the identity matrix of size k and X' is the transpose of X. +* Ik is the identity matrix of size k and X**H is the transpose of X. * kron(X, Y) is the Kronecker product between the matrices X and Y. * -* If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b +* If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b * is solved for, which is equivalent to solve for R and L in * -* A' * R + D' * L = scale * C (3) -* R * B' + L * E' = scale * -F +* A**H * R + D**H * L = scale * C (3) +* R * B**H + L * E**H = scale * -F * * This case is used to compute an estimate of Dif[(A, D), (B, E)] = * = sigma_min(Z) using reverse communicaton with CLACON. @@ -307,7 +307,7 @@ DO 80 I = 1, M DO 70 J = N, 1, -1 * -* Build 2 by 2 system Z' +* Build 2 by 2 system Z**H * Z( 1, 1 ) = CONJG( A( I, I ) ) Z( 2, 1 ) = -CONJG( B( J, J ) ) @@ -320,7 +320,7 @@ RHS( 1 ) = C( I, J ) RHS( 2 ) = F( I, J ) * -* Solve Z' * x = RHS +* Solve Z**H * x = RHS * CALL CGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) diff --git a/SRC/ctgsyl.f b/SRC/ctgsyl.f index 2d3374ca..598b35f4 100644 --- a/SRC/ctgsyl.f +++ b/SRC/ctgsyl.f @@ -39,10 +39,10 @@ * In matrix notation (1) is equivalent to solve Zx = scale*b, where Z * is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ], +* Z = [ kron(In, A) -kron(B**H, Im) ] (2) +* [ kron(In, D) -kron(E**H, Im) ], * -* Here Ix is the identity matrix of size x and X' is the conjugate +* Here Ix is the identity matrix of size x and X**H is the conjugate * transpose of X. Kron(X, Y) is the Kronecker product between the * matrices X and Y. * diff --git a/SRC/ctpcon.f b/SRC/ctpcon.f index 4cfb771d..e577a546 100644 --- a/SRC/ctpcon.f +++ b/SRC/ctpcon.f @@ -166,7 +166,7 @@ $ WORK, SCALE, RWORK, INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**H). * CALL CLATPS( UPLO, 'Conjugate transpose', DIAG, NORMIN, $ N, AP, WORK, SCALE, RWORK, INFO ) diff --git a/SRC/ctrcon.f b/SRC/ctrcon.f index 2ee42f93..0cb81c21 100644 --- a/SRC/ctrcon.f +++ b/SRC/ctrcon.f @@ -172,7 +172,7 @@ $ LDA, WORK, SCALE, RWORK, INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**H). * CALL CLATRS( UPLO, 'Conjugate transpose', DIAG, NORMIN, $ N, A, LDA, WORK, SCALE, RWORK, INFO ) diff --git a/SRC/ctrttf.f b/SRC/ctrttf.f index 4835ca4a..825cfeea 100644 --- a/SRC/ctrttf.f +++ b/SRC/ctrttf.f @@ -72,15 +72,15 @@ * 55 50 51 52 53 54 55 * * -* Let TRANSR = `N'. RFP holds AP as follows: -* For UPLO = `U' the upper trapezoid A(0:5,0:2) consists of the last +* Let TRANSR = 'N'. RFP holds AP as follows: +* For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last * three columns of AP upper. The lower triangle A(4:6,0:2) consists of * conjugate-transpose of the first three columns of AP upper. -* For UPLO = `L' the lower trapezoid A(1:6,0:2) consists of the first +* For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first * three columns of AP lower. The upper triangle A(0:2,0:2) consists of * conjugate-transpose of the last three columns of AP lower. * To denote conjugate we place -- above the element. This covers the -* case N even and TRANSR = `N'. +* case N even and TRANSR = 'N'. * * RFP A RFP A * @@ -99,7 +99,7 @@ * -- -- -- * 02 12 22 50 51 52 * -* Now let TRANSR = `C'. RFP A in both UPLO cases is just the conjugate- +* Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- * transpose of RFP A above. One therefore gets: * * @@ -125,15 +125,15 @@ * 44 40 41 42 43 44 * * -* Let TRANSR = `N'. RFP holds AP as follows: -* For UPLO = `U' the upper trapezoid A(0:4,0:2) consists of the last +* Let TRANSR = 'N'. RFP holds AP as follows: +* For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last * three columns of AP upper. The lower triangle A(3:4,0:1) consists of * conjugate-transpose of the first two columns of AP upper. -* For UPLO = `L' the lower trapezoid A(0:4,0:2) consists of the first +* For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first * three columns of AP lower. The upper triangle A(0:1,1:2) consists of * conjugate-transpose of the last two columns of AP lower. * To denote conjugate we place -- above the element. This covers the -* case N odd and TRANSR = `N'. +* case N odd and TRANSR = 'N'. * * RFP A RFP A * @@ -148,7 +148,7 @@ * -- -- * 01 11 44 40 41 42 * -* Now let TRANSR = `C'. RFP A in both UPLO cases is just the conjugate- +* Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- * transpose of RFP A above. One therefore gets: * * diff --git a/SRC/cungbr.f b/SRC/cungbr.f index 7e9a5760..5f3e0f6b 100644 --- a/SRC/cungbr.f +++ b/SRC/cungbr.f @@ -202,7 +202,7 @@ END IF ELSE * -* Form P', determined by a call to CGEBRD to reduce a k-by-n +* Form P**H, determined by a call to CGEBRD to reduce a k-by-n * matrix * IF( K.LT.N ) THEN @@ -216,7 +216,7 @@ * If k >= n, assume m = n * * Shift the vectors which define the elementary reflectors one -* row downward, and set the first row and column of P' to +* row downward, and set the first row and column of P**H to * those of the unit matrix * A( 1, 1 ) = ONE @@ -231,7 +231,7 @@ 60 CONTINUE IF( N.GT.1 ) THEN * -* Form P'(2:n,2:n) +* Form P**H(2:n,2:n) * CALL CUNGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK, $ LWORK, IINFO ) diff --git a/SRC/dbdsdc.f b/SRC/dbdsdc.f index 1904a61a..af08e139 100644 --- a/SRC/dbdsdc.f +++ b/SRC/dbdsdc.f @@ -77,7 +77,7 @@ * * VT (output) DOUBLE PRECISION array, dimension (LDVT,N) * If COMPQ = 'I', then: -* On exit, if INFO = 0, VT' contains the right singular +* On exit, if INFO = 0, VT**T contains the right singular * vectors of the bidiagonal matrix. * For other values of COMPQ, VT is not referenced. * diff --git a/SRC/dgbtrs.f b/SRC/dgbtrs.f index 3b0571e4..8fa729ae 100644 --- a/SRC/dgbtrs.f +++ b/SRC/dgbtrs.f @@ -19,7 +19,7 @@ * ======= * * DGBTRS solves a system of linear equations -* A * X = B or A' * X = B +* A * X = B or A**T * X = B * with a general band matrix A using the LU factorization computed * by DGBTRF. * @@ -29,8 +29,8 @@ * TRANS (input) CHARACTER*1 * Specifies the form of the system of equations. * = 'N': A * X = B (No transpose) -* = 'T': A'* X = B (Transpose) -* = 'C': A'* X = B (Conjugate transpose = Transpose) +* = 'T': A**T* X = B (Transpose) +* = 'C': A**T* X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The order of the matrix A. N >= 0. @@ -157,17 +157,17 @@ * ELSE * -* Solve A'*X = B. +* Solve A**T*X = B. * DO 30 I = 1, NRHS * -* Solve U'*X = B, overwriting B with X. +* Solve U**T*X = B, overwriting B with X. * CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU, AB, $ LDAB, B( 1, I ), 1 ) 30 CONTINUE * -* Solve L'*X = B, overwriting B with X. +* Solve L**T*X = B, overwriting B with X. * IF( LNOTI ) THEN DO 40 J = N - 1, 1, -1 diff --git a/SRC/dgebd2.f b/SRC/dgebd2.f index 5807773b..93e22627 100644 --- a/SRC/dgebd2.f +++ b/SRC/dgebd2.f @@ -17,7 +17,7 @@ * ======= * * DGEBD2 reduces a real general m by n matrix A to upper or lower -* bidiagonal form B by an orthogonal transformation: Q' * A * P = B. +* bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. * * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. * diff --git a/SRC/dgels.f b/SRC/dgels.f index 53d3fc6b..49a423cc 100644 --- a/SRC/dgels.f +++ b/SRC/dgels.f @@ -298,9 +298,9 @@ * ELSE * -* Overdetermined system of equations A' * X = B +* Overdetermined system of equations A**T * X = B * -* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) +* B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS) * CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS, $ A, LDA, B, LDB, INFO ) @@ -371,7 +371,7 @@ * ELSE * -* overdetermined system min || A' * X - B || +* overdetermined system min || A**T * X - B || * * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) * @@ -381,7 +381,7 @@ * * workspace at least NRHS, optimally NRHS*NB * -* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) +* B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS) * CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS, $ A, LDA, B, LDB, INFO ) diff --git a/SRC/dggsvd.f b/SRC/dggsvd.f index 9b5bb017..e235cc7a 100644 --- a/SRC/dggsvd.f +++ b/SRC/dggsvd.f @@ -128,7 +128,7 @@ * L (output) INTEGER * On exit, K and L specify the dimension of the subblocks * described in the Purpose section. -* K + L = effective numerical rank of (A',B')**T. +* K + L = effective numerical rank of (A**T,B**T)**T. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. diff --git a/SRC/dgttrs.f b/SRC/dgttrs.f index 1b83afd8..2373fb4a 100644 --- a/SRC/dgttrs.f +++ b/SRC/dgttrs.f @@ -19,7 +19,7 @@ * ======= * * DGTTRS solves one of the systems of equations -* A*X = B or A'*X = B, +* A*X = B or A**T*X = B, * with a tridiagonal matrix A using the LU factorization computed * by DGTTRF. * @@ -29,8 +29,8 @@ * TRANS (input) CHARACTER*1 * Specifies the form of the system of equations. * = 'N': A * X = B (No transpose) -* = 'T': A'* X = B (Transpose) -* = 'C': A'* X = B (Conjugate transpose = Transpose) +* = 'T': A**T* X = B (Transpose) +* = 'C': A**T* X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The order of the matrix A. diff --git a/SRC/dgtts2.f b/SRC/dgtts2.f index 4e3bc5bc..98a13c17 100644 --- a/SRC/dgtts2.f +++ b/SRC/dgtts2.f @@ -17,7 +17,7 @@ * ======= * * DGTTS2 solves one of the systems of equations -* A*X = B or A'*X = B, +* A*X = B or A**T*X = B, * with a tridiagonal matrix A using the LU factorization computed * by DGTTRF. * @@ -27,8 +27,8 @@ * ITRANS (input) INTEGER * Specifies the form of the system of equations. * = 0: A * X = B (No transpose) -* = 1: A'* X = B (Transpose) -* = 2: A'* X = B (Conjugate transpose = Transpose) +* = 1: A**T* X = B (Transpose) +* = 2: A**T* X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The order of the matrix A. @@ -138,11 +138,11 @@ END IF ELSE * -* Solve A' * X = B. +* Solve A**T * X = B. * IF( NRHS.LE.1 ) THEN * -* Solve U'*x = b. +* Solve U**T*x = b. * J = 1 70 CONTINUE @@ -154,7 +154,7 @@ $ B( I-2, J ) ) / D( I ) 80 CONTINUE * -* Solve L'*x = b. +* Solve L**T*x = b. * DO 90 I = N - 1, 1, -1 IP = IPIV( I ) @@ -170,7 +170,7 @@ ELSE DO 120 J = 1, NRHS * -* Solve U'*x = b. +* Solve U**T*x = b. * B( 1, J ) = B( 1, J ) / D( 1 ) IF( N.GT.1 ) diff --git a/SRC/dla_gbrcond.f b/SRC/dla_gbrcond.f index f2f02b56..7875adb9 100644 --- a/SRC/dla_gbrcond.f +++ b/SRC/dla_gbrcond.f @@ -226,7 +226,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/dla_gercond.f b/SRC/dla_gercond.f index 5e414037..85ae26b5 100644 --- a/SRC/dla_gercond.f +++ b/SRC/dla_gercond.f @@ -208,7 +208,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/dla_porcond.f b/SRC/dla_porcond.f index b32afd0d..9eb572f2 100644 --- a/SRC/dla_porcond.f +++ b/SRC/dla_porcond.f @@ -213,7 +213,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/dla_syrcond.f b/SRC/dla_syrcond.f index c61f6b0e..fd39a3c5 100644 --- a/SRC/dla_syrcond.f +++ b/SRC/dla_syrcond.f @@ -219,7 +219,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/dlacn2.f b/SRC/dlacn2.f index 26b8f3c7..f3e09d3d 100644 --- a/SRC/dlacn2.f +++ b/SRC/dlacn2.f @@ -33,7 +33,7 @@ * X (input/output) DOUBLE PRECISION array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, +* A**T * X, if KASE=2, * and DLACN2 must be re-called with all the other parameters * unchanged. * @@ -47,7 +47,7 @@ * KASE (input/output) INTEGER * On the initial call to DLACN2, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**T * X. * On the final return from DLACN2, KASE will again be 0. * * ISAVE (input/output) INTEGER array, dimension (3) diff --git a/SRC/dlacon.f b/SRC/dlacon.f index 98c56831..3b050e15 100644 --- a/SRC/dlacon.f +++ b/SRC/dlacon.f @@ -33,7 +33,7 @@ * X (input/output) DOUBLE PRECISION array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, +* A**T * X, if KASE=2, * and DLACON must be re-called with all the other parameters * unchanged. * @@ -47,7 +47,7 @@ * KASE (input/output) INTEGER * On the initial call to DLACON, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**T * X. * On the final return from DLACON, KASE will again be 0. * * Further Details diff --git a/SRC/dlaed1.f b/SRC/dlaed1.f index ad45f6ef..df64289c 100644 --- a/SRC/dlaed1.f +++ b/SRC/dlaed1.f @@ -26,9 +26,9 @@ * of a full symmetric matrix (which was reduced to tridiagonal form) * are desired. * -* T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out) +* T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) * -* where Z = Q'u, u is a vector of length N with ones in the +* where Z = Q**T*u, u is a vector of length N with ones in the * CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. * * The eigenvectors of the original matrix are stored in Q, and the diff --git a/SRC/dlaein.f b/SRC/dlaein.f index 991b7f23..1149895a 100644 --- a/SRC/dlaein.f +++ b/SRC/dlaein.f @@ -427,7 +427,7 @@ VCRIT = BIGNUM * * Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector, -* or U'*(xr,xi) = scale*(vr,vi) for a left eigenvector, +* or U**T*(xr,xi) = scale*(vr,vi) for a left eigenvector, * overwriting (xr,xi) on (vr,vi). * DO 250 I = I1, I2, I3 diff --git a/SRC/dlags2.f b/SRC/dlags2.f index d8541123..5b0e5ab1 100644 --- a/SRC/dlags2.f +++ b/SRC/dlags2.f @@ -37,7 +37,7 @@ * U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) * ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) * -* Z' denotes the transpose of Z. +* Z**T denotes the transpose of Z. * * * Arguments diff --git a/SRC/dlagv2.f b/SRC/dlagv2.f index 7bbff22e..1e00144a 100644 --- a/SRC/dlagv2.f +++ b/SRC/dlagv2.f @@ -242,7 +242,7 @@ CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR, $ CSR, SNL, CSL ) * -* Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and +* Form (A,B) := Q(A,B)Z**T where Q is left rotation matrix and * Z is right rotation matrix computed from DLASV2 * CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) diff --git a/SRC/dlahr2.f b/SRC/dlahr2.f index 87bf4220..3ea96d13 100644 --- a/SRC/dlahr2.f +++ b/SRC/dlahr2.f @@ -149,7 +149,7 @@ CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY, $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 ) * -* Apply I - V * T' * V**T to this column (call it b) from the +* Apply I - V * T**T * V**T to this column (call it b) from the * left, using the last column of T as workspace * * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) @@ -157,7 +157,7 @@ * * where V1 is unit lower triangular * -* w := V1' * b1 +* w := V1**T * b1 * CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL DTRMV( 'Lower', 'Transpose', 'UNIT', diff --git a/SRC/dlaic1.f b/SRC/dlaic1.f index b62a69d6..5873e80f 100644 --- a/SRC/dlaic1.f +++ b/SRC/dlaic1.f @@ -27,15 +27,15 @@ * [ s*x ] * xhat = [ c ] * is an approximate singular vector of -* [ L 0 ] -* Lhat = [ w' gamma ] +* [ L 0 ] +* Lhat = [ w**T gamma ] * in the sense that * twonorm(Lhat*xhat) = sestpr. * * Depending on JOB, an estimate for the largest or smallest singular * value is computed. * -* Note that [s c]' and sestpr**2 is an eigenpair of the system +* Note that [s c]**T and sestpr**2 is an eigenpair of the system * * diag(sest*sest, 0) + [alpha gamma] * [ alpha ] * [ gamma ] diff --git a/SRC/dlaln2.f b/SRC/dlaln2.f index 4e38e4d0..59ea7d42 100644 --- a/SRC/dlaln2.f +++ b/SRC/dlaln2.f @@ -19,8 +19,8 @@ * ======= * * DLALN2 solves a system of the form (ca A - w D ) X = s B -* or (ca A' - w D) X = s B with possible scaling ("s") and -* perturbation of A. (A' means A-transpose.) +* or (ca A**T - w D) X = s B with possible scaling ("s") and +* perturbation of A. (A**T means A-transpose.) * * A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA * real diagonal matrix, w is a real or complex value, and X and B are @@ -257,7 +257,7 @@ * * 2x2 System * -* Compute the real part of C = ca A - w D (or ca A' - w D ) +* Compute the real part of C = ca A - w D (or ca A**T - w D ) * CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1 CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2 diff --git a/SRC/dlalsa.f b/SRC/dlalsa.f index 83751a6d..e5e2f7ae 100644 --- a/SRC/dlalsa.f +++ b/SRC/dlalsa.f @@ -81,7 +81,7 @@ * POLES, GIVNUM, and Z. * * VT (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ). -* On entry, VT' contains the right singular vector matrices of +* On entry, VT**T contains the right singular vector matrices of * all subproblems at the bottom level. * * K (input) INTEGER array, dimension ( N ). diff --git a/SRC/dlansf.f b/SRC/dlansf.f index 6ba669b7..21d63c7f 100644 --- a/SRC/dlansf.f +++ b/SRC/dlansf.f @@ -724,7 +724,7 @@ ELSE * A is xpose IF( ILU.EQ.0 ) THEN -* A' is upper +* A**T is upper DO J = 1, K - 2 CALL DLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE, S ) * U at A(0,k) @@ -745,7 +745,7 @@ CALL DLASSQ( K, A( 0+( K-1 )*LDA ), LDA+1, SCALE, S ) * tri L at A(0,k-1) ELSE -* A' is lower +* A**T is lower DO J = 1, K - 1 CALL DLASSQ( J, A( 0+J*LDA ), 1, SCALE, S ) * U at A(0,0) @@ -806,7 +806,7 @@ ELSE * A is xpose IF( ILU.EQ.0 ) THEN -* A' is upper +* A**T is upper DO J = 1, K - 1 CALL DLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE, S ) * U at A(0,k+1) @@ -827,7 +827,7 @@ CALL DLASSQ( K, A( 0+K*LDA ), LDA+1, SCALE, S ) * tri L at A(0,k) ELSE -* A' is lower +* A**T is lower DO J = 1, K - 1 CALL DLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE, S ) * U at A(0,1) diff --git a/SRC/dlaqps.f b/SRC/dlaqps.f index 150b0812..161a7d8e 100644 --- a/SRC/dlaqps.f +++ b/SRC/dlaqps.f @@ -76,7 +76,7 @@ * Auxiliar vector. * * F (input/output) DOUBLE PRECISION array, dimension (LDF,NB) -* Matrix F' = L*Y'*A. +* Matrix F**T = L*Y**T*A. * * LDF (input) INTEGER * The leading dimension of the array F. LDF >= max(1,N). diff --git a/SRC/dlaqr5.f b/SRC/dlaqr5.f index 06381e1c..0615eaf8 100644 --- a/SRC/dlaqr5.f +++ b/SRC/dlaqr5.f @@ -642,14 +642,14 @@ CALL DLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ), $ LDH, WH( KZS+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**T ==== * CALL DLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH ) CALL DTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE, $ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ), $ LDWH ) * -* ==== Multiply top of H by U11' ==== +* ==== Multiply top of H by U11**T ==== * CALL DGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU, $ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH ) @@ -659,7 +659,7 @@ CALL DLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH, $ WH( I2+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**T ==== * CALL DTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE, $ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH ) diff --git a/SRC/dlaqtr.f b/SRC/dlaqtr.f index 18c42f46..07c8346d 100644 --- a/SRC/dlaqtr.f +++ b/SRC/dlaqtr.f @@ -36,7 +36,7 @@ * [ . ] * [ w ] * -* op(A) = A or A', A' denotes the conjugate transpose of +* op(A) = A or A**T, A**T denotes the transpose of * matrix A. * * On input, X = [ c ]. On output, X = [ p ]. @@ -290,7 +290,7 @@ * ELSE * -* Solve T'*p = scale*c +* Solve T**T*p = scale*c * JNEXT = 1 DO 40 J = 1, N diff --git a/SRC/dlarfb.f b/SRC/dlarfb.f index 024a0b46..c2b63e07 100644 --- a/SRC/dlarfb.f +++ b/SRC/dlarfb.f @@ -286,7 +286,7 @@ LASTV = MAX( K, ILADLR( M, K, V, LDV ) ) LASTC = ILADLC( LASTV, N, C, LDC ) * -* W := C' * V = (C1**T * V1 + C2**T * V2) (stored in WORK) +* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK) * * W := C2**T * @@ -373,7 +373,7 @@ CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', $ LASTC, K, ONE, T, LDT, WORK, LDWORK ) * -* C := C - W * V' +* C := C - W * V**T * IF( LASTV.GT.K ) THEN * diff --git a/SRC/dlartgs.f b/SRC/dlartgs.f index 87f4263d..d95d5562 100644 --- a/SRC/dlartgs.f +++ b/SRC/dlartgs.f @@ -61,7 +61,7 @@ * THRESH = DLAMCH('E') * -* Compute the first column of B'*B - SIGMA^2*I, up to a scale +* Compute the first column of B**T*B - SIGMA^2*I, up to a scale * factor. * IF( (SIGMA .EQ. ZERO .AND. ABS(X) .LT. THRESH) .OR. diff --git a/SRC/dlarzb.f b/SRC/dlarzb.f index fbe5a9a0..a7a1eeac 100644 --- a/SRC/dlarzb.f +++ b/SRC/dlarzb.f @@ -155,7 +155,7 @@ $ CALL DGEMM( 'Transpose', 'Transpose', N, K, L, ONE, $ C( M-L+1, 1 ), LDC, V, LDV, ONE, WORK, LDWORK ) * -* W( 1:n, 1:k ) = W( 1:n, 1:k ) * T' or W( 1:m, 1:k ) * T +* W( 1:n, 1:k ) = W( 1:n, 1:k ) * T**T or W( 1:m, 1:k ) * T * CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, ONE, T, $ LDT, WORK, LDWORK ) diff --git a/SRC/dlarzt.f b/SRC/dlarzt.f index dbbe29c7..f0246d52 100644 --- a/SRC/dlarzt.f +++ b/SRC/dlarzt.f @@ -27,12 +27,12 @@ * If STOREV = 'C', the vector which defines the elementary reflector * H(i) is stored in the i-th column of the array V, and * -* H = I - V * T * V' +* H = I - V * T * V**T * * If STOREV = 'R', the vector which defines the elementary reflector * H(i) is stored in the i-th row of the array V, and * -* H = I - V' * T * V +* H = I - V**T * T * V * * Currently, only STOREV = 'R' and DIRECT = 'B' are supported. * diff --git a/SRC/dlasd0.f b/SRC/dlasd0.f index 7fcc49ea..2ad48270 100644 --- a/SRC/dlasd0.f +++ b/SRC/dlasd0.f @@ -55,7 +55,7 @@ * On entry, leading dimension of U. * * VT (output) DOUBLE PRECISION array, dimension at least (LDVT, M) -* On exit, VT' contains the right singular vectors. +* On exit, VT**T contains the right singular vectors. * * LDVT (input) INTEGER * On entry, leading dimension of VT. diff --git a/SRC/dlasd1.f b/SRC/dlasd1.f index 797cc0c0..bf73e9ca 100644 --- a/SRC/dlasd1.f +++ b/SRC/dlasd1.f @@ -26,13 +26,13 @@ * * DLASD1 computes the SVD as follows: * -* ( D1(in) 0 0 0 ) -* B = U(in) * ( Z1' a Z2' b ) * VT(in) -* ( 0 0 D2(in) 0 ) +* ( D1(in) 0 0 0 ) +* B = U(in) * ( Z1**T a Z2**T b ) * VT(in) +* ( 0 0 D2(in) 0 ) * * = U(out) * ( D(out) 0) * VT(out) * -* where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M +* where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M * with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros * elsewhere; and the entry b is empty if SQRE = 0. * @@ -101,7 +101,7 @@ * On entry VT(1:NL+1, 1:NL+1)**T contains the right singular * vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains * the right singular vectors of the lower block. On exit -* VT' contains the right singular vectors of the +* VT**T contains the right singular vectors of the * bidiagonal matrix. * * LDVT (input) INTEGER diff --git a/SRC/dlasd2.f b/SRC/dlasd2.f index caeaff06..7ff9066e 100644 --- a/SRC/dlasd2.f +++ b/SRC/dlasd2.f @@ -79,10 +79,10 @@ * The leading dimension of the array U. LDU >= N. * * VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M) -* On entry VT' contains the right singular vectors of two +* On entry VT**T contains the right singular vectors of two * submatrices in the two square blocks with corners at (1,1), * (NL+1, NL+1), and (NL+2, NL+2), (M,M). -* On exit VT' contains the trailing (N-K) updated right singular +* On exit VT**T contains the trailing (N-K) updated right singular * vectors (those which were deflated) in its last N-K columns. * In case SQRE =1, the last row of VT spans the right null * space. @@ -107,7 +107,7 @@ * The leading dimension of the array U2. LDU2 >= N. * * VT2 (output) DOUBLE PRECISION array, dimension(LDVT2,N) -* VT2' contains a copy of the first K right singular vectors +* VT2**T contains a copy of the first K right singular vectors * which will be used by DLASD3 in a matrix multiply (DGEMM) to * solve for the new right singular vectors. VT2 is arranged into * three blocks. The first block contains a row that corresponds diff --git a/SRC/dlasd3.f b/SRC/dlasd3.f index cbe5cd98..5f3111c7 100644 --- a/SRC/dlasd3.f +++ b/SRC/dlasd3.f @@ -84,14 +84,14 @@ * The leading dimension of the array U2. LDU2 >= N. * * VT (output) DOUBLE PRECISION array, dimension (LDVT, M) -* The last M - K columns of VT' contain the deflated +* The last M - K columns of VT**T contain the deflated * right singular vectors. * * LDVT (input) INTEGER * The leading dimension of the array VT. LDVT >= N. * * VT2 (input/output) DOUBLE PRECISION array, dimension (LDVT2, N) -* The first K columns of VT2' contain the non-deflated +* The first K columns of VT2**T contain the non-deflated * right singular vectors for the split problem. * * LDVT2 (input) INTEGER diff --git a/SRC/dlasd6.f b/SRC/dlasd6.f index 32aad149..d86465d5 100644 --- a/SRC/dlasd6.f +++ b/SRC/dlasd6.f @@ -34,13 +34,13 @@ * * DLASD6 computes the SVD as follows: * -* ( D1(in) 0 0 0 ) -* B = U(in) * ( Z1' a Z2' b ) * VT(in) -* ( 0 0 D2(in) 0 ) +* ( D1(in) 0 0 0 ) +* B = U(in) * ( Z1**T a Z2**T b ) * VT(in) +* ( 0 0 D2(in) 0 ) * * = U(out) * ( D(out) 0) * VT(out) * -* where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M +* where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M * with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros * elsewhere; and the entry b is empty if SQRE = 0. * diff --git a/SRC/dlasda.f b/SRC/dlasda.f index b1f12207..f852dfc9 100644 --- a/SRC/dlasda.f +++ b/SRC/dlasda.f @@ -75,7 +75,7 @@ * * VT (output) DOUBLE PRECISION array, * dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced -* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right +* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right * singular vector matrices of all subproblems at the bottom * level. * diff --git a/SRC/dlasdq.f b/SRC/dlasdq.f index 4bd85263..92764640 100644 --- a/SRC/dlasdq.f +++ b/SRC/dlasdq.f @@ -22,12 +22,12 @@ * (upper or lower) bidiagonal matrix with diagonal D and offdiagonal * E, accumulating the transformations if desired. Letting B denote * the input bidiagonal matrix, the algorithm computes orthogonal -* matrices Q and P such that B = Q * S * P' (P' denotes the transpose +* matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose * of P). The singular values S are overwritten on D. * * The input matrix U is changed to U * Q if desired. -* The input matrix VT is changed to P' * VT if desired. -* The input matrix C is changed to Q' * C if desired. +* The input matrix VT is changed to P**T * VT if desired. +* The input matrix C is changed to Q**T * C if desired. * * See "Computing Small Singular Values of Bidiagonal Matrices With * Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, @@ -84,7 +84,7 @@ * * VT (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT) * On entry, contains a matrix which on exit has been -* premultiplied by P', dimension N-by-NCVT if SQRE = 0 +* premultiplied by P**T, dimension N-by-NCVT if SQRE = 0 * and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0). * * LDVT (input) INTEGER @@ -104,7 +104,7 @@ * * C (input/output) DOUBLE PRECISION array, dimension (LDC, NCC) * On entry, contains an N-by-NCC matrix which on exit -* has been premultiplied by Q' dimension N-by-NCC if SQRE = 0 +* has been premultiplied by Q**T dimension N-by-NCC if SQRE = 0 * and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0). * * LDC (input) INTEGER diff --git a/SRC/dlasr.f b/SRC/dlasr.f index 4c0d868b..55ab2e37 100644 --- a/SRC/dlasr.f +++ b/SRC/dlasr.f @@ -272,7 +272,7 @@ END IF ELSE IF( LSAME( SIDE, 'R' ) ) THEN * -* Form A * P' +* Form A * P**T * IF( LSAME( PIVOT, 'V' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN diff --git a/SRC/dlasy2.f b/SRC/dlasy2.f index db53ca4a..d88ddb1d 100644 --- a/SRC/dlasy2.f +++ b/SRC/dlasy2.f @@ -24,7 +24,7 @@ * op(TL)*X + ISGN*X*op(TR) = SCALE*B, * * where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or -* -1. op(T) = T or T', where T' denotes the transpose of T. +* -1. op(T) = T or T**T, where T**T denotes the transpose of T. * * Arguments * ========= @@ -32,12 +32,12 @@ * LTRANL (input) LOGICAL * On entry, LTRANL specifies the op(TL): * = .FALSE., op(TL) = TL, -* = .TRUE., op(TL) = TL'. +* = .TRUE., op(TL) = TL**T. * * LTRANR (input) LOGICAL * On entry, LTRANR specifies the op(TR): * = .FALSE., op(TR) = TR, -* = .TRUE., op(TR) = TR'. +* = .TRUE., op(TR) = TR**T. * * ISGN (input) INTEGER * On entry, ISGN specifies the sign of the equation diff --git a/SRC/dlatbs.f b/SRC/dlatbs.f index ded346ee..2c991d3d 100644 --- a/SRC/dlatbs.f +++ b/SRC/dlatbs.f @@ -20,10 +20,10 @@ * * DLATBS solves one of the triangular systems * -* A *x = s*b or A'*x = s*b +* A *x = s*b or A**T*x = s*b * * with scaling to prevent overflow, where A is an upper or lower -* triangular band matrix. Here A' denotes the transpose of A, x and b +* triangular band matrix. Here A**T denotes the transpose of A, x and b * are n-element vectors, and s is a scaling factor, usually less than * or equal to 1, chosen so that the components of x will be less than * the overflow threshold. If the unscaled problem will not cause @@ -42,8 +42,8 @@ * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': Solve A * x = s*b (No transpose) -* = 'T': Solve A'* x = s*b (Transpose) -* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) +* = 'T': Solve A**T* x = s*b (Transpose) +* = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. @@ -79,7 +79,7 @@ * * SCALE (output) DOUBLE PRECISION * The scaling factor s for the triangular system -* A * x = s*b or A'* x = s*b. +* A * x = s*b or A**T* x = s*b. * If SCALE = 0, the matrix A is singular or badly scaled, and * the vector x is an exact or approximate solution to A*x = 0. * @@ -145,15 +145,15 @@ * prevent overflow, but if the bound overflows, x is set to 0, x(j) to * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. * -* Similarly, a row-wise scheme is used to solve A'*x = b. The basic +* Similarly, a row-wise scheme is used to solve A**T*x = b. The basic * algorithm for A upper triangular is * * for j = 1, ..., n -* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) +* x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) * end * * We simultaneously compute two bounds -* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j +* G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j * M(j) = bound on x(i), 1<=i<=j * * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we @@ -356,7 +356,7 @@ * ELSE * -* Compute the growth in A' * x = b. +* Compute the growth in A**T * x = b. * IF( UPPER ) THEN JFIRST = 1 @@ -569,7 +569,7 @@ * ELSE * -* Solve A' * x = b +* Solve A**T * x = b * DO 160 J = JFIRST, JLAST, JINC * @@ -688,7 +688,7 @@ ELSE * * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and -* scale = 0, and compute a solution to A'*x = 0. +* scale = 0, and compute a solution to A**T*x = 0. * DO 140 I = 1, N X( I ) = ZERO diff --git a/SRC/dlatps.f b/SRC/dlatps.f index 02671dc6..667d517a 100644 --- a/SRC/dlatps.f +++ b/SRC/dlatps.f @@ -20,10 +20,10 @@ * * DLATPS solves one of the triangular systems * -* A *x = s*b or A'*x = s*b +* A *x = s*b or A**T*x = s*b * * with scaling to prevent overflow, where A is an upper or lower -* triangular matrix stored in packed form. Here A' denotes the +* triangular matrix stored in packed form. Here A**T denotes the * transpose of A, x and b are n-element vectors, and s is a scaling * factor, usually less than or equal to 1, chosen so that the * components of x will be less than the overflow threshold. If the @@ -42,8 +42,8 @@ * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': Solve A * x = s*b (No transpose) -* = 'T': Solve A'* x = s*b (Transpose) -* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) +* = 'T': Solve A**T* x = s*b (Transpose) +* = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. @@ -72,7 +72,7 @@ * * SCALE (output) DOUBLE PRECISION * The scaling factor s for the triangular system -* A * x = s*b or A'* x = s*b. +* A * x = s*b or A**T* x = s*b. * If SCALE = 0, the matrix A is singular or badly scaled, and * the vector x is an exact or approximate solution to A*x = 0. * @@ -138,15 +138,15 @@ * prevent overflow, but if the bound overflows, x is set to 0, x(j) to * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. * -* Similarly, a row-wise scheme is used to solve A'*x = b. The basic +* Similarly, a row-wise scheme is used to solve A**T*x = b. The basic * algorithm for A upper triangular is * * for j = 1, ..., n -* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) +* x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) * end * * We simultaneously compute two bounds -* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j +* G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j * M(j) = bound on x(i), 1<=i<=j * * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we @@ -346,7 +346,7 @@ * ELSE * -* Compute the growth in A' * x = b. +* Compute the growth in A**T * x = b. * IF( UPPER ) THEN JFIRST = 1 @@ -561,7 +561,7 @@ * ELSE * -* Solve A' * x = b +* Solve A**T * x = b * IP = JFIRST*( JFIRST+1 ) / 2 JLEN = 1 @@ -675,7 +675,7 @@ ELSE * * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and -* scale = 0, and compute a solution to A'*x = 0. +* scale = 0, and compute a solution to A**T*x = 0. * DO 140 I = 1, N X( I ) = ZERO diff --git a/SRC/dlatrs.f b/SRC/dlatrs.f index de70c1f5..b8fd5c31 100644 --- a/SRC/dlatrs.f +++ b/SRC/dlatrs.f @@ -20,10 +20,10 @@ * * DLATRS solves one of the triangular systems * -* A *x = s*b or A'*x = s*b +* A *x = s*b or A**T *x = s*b * * with scaling to prevent overflow. Here A is an upper or lower -* triangular matrix, A' denotes the transpose of A, x and b are +* triangular matrix, A**T denotes the transpose of A, x and b are * n-element vectors, and s is a scaling factor, usually less than * or equal to 1, chosen so that the components of x will be less than * the overflow threshold. If the unscaled problem will not cause @@ -42,8 +42,8 @@ * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': Solve A * x = s*b (No transpose) -* = 'T': Solve A'* x = s*b (Transpose) -* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) +* = 'T': Solve A**T* x = s*b (Transpose) +* = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. @@ -78,7 +78,7 @@ * * SCALE (output) DOUBLE PRECISION * The scaling factor s for the triangular system -* A * x = s*b or A'* x = s*b. +* A * x = s*b or A**T* x = s*b. * If SCALE = 0, the matrix A is singular or badly scaled, and * the vector x is an exact or approximate solution to A*x = 0. * @@ -144,15 +144,15 @@ * prevent overflow, but if the bound overflows, x is set to 0, x(j) to * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. * -* Similarly, a row-wise scheme is used to solve A'*x = b. The basic +* Similarly, a row-wise scheme is used to solve A**T*x = b. The basic * algorithm for A upper triangular is * * for j = 1, ..., n -* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) +* x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) * end * * We simultaneously compute two bounds -* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j +* G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j * M(j) = bound on x(i), 1<=i<=j * * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we @@ -346,7 +346,7 @@ * ELSE * -* Compute the growth in A' * x = b. +* Compute the growth in A**T * x = b. * IF( UPPER ) THEN JFIRST = 1 @@ -554,7 +554,7 @@ * ELSE * -* Solve A' * x = b +* Solve A**T * x = b * DO 160 J = JFIRST, JLAST, JINC * @@ -666,7 +666,7 @@ ELSE * * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and -* scale = 0, and compute a solution to A'*x = 0. +* scale = 0, and compute a solution to A**T*x = 0. * DO 140 I = 1, N X( I ) = ZERO diff --git a/SRC/dorgbr.f b/SRC/dorgbr.f index 0e7e884f..2bb421a4 100644 --- a/SRC/dorgbr.f +++ b/SRC/dorgbr.f @@ -201,7 +201,7 @@ END IF ELSE * -* Form P', determined by a call to DGEBRD to reduce a k-by-n +* Form P**T, determined by a call to DGEBRD to reduce a k-by-n * matrix * IF( K.LT.N ) THEN @@ -215,7 +215,7 @@ * If k >= n, assume m = n * * Shift the vectors which define the elementary reflectors one -* row downward, and set the first row and column of P' to +* row downward, and set the first row and column of P**T to * those of the unit matrix * A( 1, 1 ) = ONE @@ -230,7 +230,7 @@ 60 CONTINUE IF( N.GT.1 ) THEN * -* Form P'(2:n,2:n) +* Form P**T(2:n,2:n) * CALL DORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK, $ LWORK, IINFO ) diff --git a/SRC/dpbrfs.f b/SRC/dpbrfs.f index f0eaa42e..91637b88 100644 --- a/SRC/dpbrfs.f +++ b/SRC/dpbrfs.f @@ -304,7 +304,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, $ INFO ) diff --git a/SRC/dporfs.f b/SRC/dporfs.f index cf2cb7e5..de8df341 100644 --- a/SRC/dporfs.f +++ b/SRC/dporfs.f @@ -296,7 +296,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) DO 110 I = 1, N diff --git a/SRC/dpprfs.f b/SRC/dpprfs.f index 26fbabc3..9efb55c9 100644 --- a/SRC/dpprfs.f +++ b/SRC/dpprfs.f @@ -293,7 +293,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) DO 110 I = 1, N diff --git a/SRC/dpstf2.f b/SRC/dpstf2.f index 35279a06..8d184b65 100644 --- a/SRC/dpstf2.f +++ b/SRC/dpstf2.f @@ -21,8 +21,8 @@ * pivoting of a real symmetric positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**T * U , if UPLO = 'U', +* P**T * A * P = L * L**T, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -161,7 +161,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**T * U * DO 130 J = 1, N * @@ -224,7 +224,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**T * DO 150 J = 1, N * diff --git a/SRC/dpstrf.f b/SRC/dpstrf.f index b253b969..24da7f9a 100644 --- a/SRC/dpstrf.f +++ b/SRC/dpstrf.f @@ -21,8 +21,8 @@ * pivoting of a real symmetric positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**T * U , if UPLO = 'U', +* P**T * A * P = L * L**T, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -170,7 +170,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**T * U * DO 140 K = 1, N, NB * @@ -257,7 +257,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**T * DO 180 K = 1, N, NB * diff --git a/SRC/dptcon.f b/SRC/dptcon.f index a0d162c2..dfb7a7b4 100644 --- a/SRC/dptcon.f +++ b/SRC/dptcon.f @@ -117,7 +117,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**T. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T. * * Solve M(L) * x = e. * diff --git a/SRC/dptrfs.f b/SRC/dptrfs.f index 2f7dfd93..99f5ebe5 100644 --- a/SRC/dptrfs.f +++ b/SRC/dptrfs.f @@ -263,7 +263,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**T. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T. * * Solve M(L) * x = e. * diff --git a/SRC/dsfrk.f b/SRC/dsfrk.f index 63a42a0a..1b136f85 100644 --- a/SRC/dsfrk.f +++ b/SRC/dsfrk.f @@ -26,11 +26,11 @@ * * DSFRK performs one of the symmetric rank--k operations * -* C := alpha*A*A' + beta*C, +* C := alpha*A*A**T + beta*C, * * or * -* C := alpha*A'*A + beta*C, +* C := alpha*A**T*A + beta*C, * * where alpha and beta are real scalars, C is an n--by--n symmetric * matrix and A is an n--by--k matrix in the first case and a k--by--n @@ -60,9 +60,9 @@ * On entry, TRANS specifies the operation to be performed as * follows: * -* TRANS = 'N' or 'n' C := alpha*A*A' + beta*C. +* TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C. * -* TRANS = 'T' or 't' C := alpha*A'*A + beta*C. +* TRANS = 'T' or 't' C := alpha*A**T*A + beta*C. * * Unchanged on exit. * diff --git a/SRC/dsprfs.f b/SRC/dsprfs.f index 8918f2c1..1eefde58 100644 --- a/SRC/dsprfs.f +++ b/SRC/dsprfs.f @@ -298,7 +298,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, $ INFO ) diff --git a/SRC/dsptrd.f b/SRC/dsptrd.f index 331dd2af..56b436d8 100644 --- a/SRC/dsptrd.f +++ b/SRC/dsptrd.f @@ -208,7 +208,7 @@ CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**T +* A := A - v * w**T - w * v**T * CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, $ AP( I1I1 ) ) diff --git a/SRC/dsyrfs.f b/SRC/dsyrfs.f index 915c85ab..59ad546b 100644 --- a/SRC/dsyrfs.f +++ b/SRC/dsyrfs.f @@ -302,7 +302,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, $ INFO ) diff --git a/SRC/dsytd2.f b/SRC/dsytd2.f index da799b95..eeede7a7 100644 --- a/SRC/dsytd2.f +++ b/SRC/dsytd2.f @@ -229,7 +229,7 @@ CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**T +* A := A - v * w**T - w * v**T * CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, $ A( I+1, I+1 ), LDA ) diff --git a/SRC/dsytrd.f b/SRC/dsytrd.f index 1bc8fec6..0622ce67 100644 --- a/SRC/dsytrd.f +++ b/SRC/dsytrd.f @@ -235,7 +235,7 @@ $ LDWORK ) * * Update the unreduced submatrix A(1:i-1,1:i-1), using an -* update of the form: A := A - V*W' - W*V**T +* update of the form: A := A - V*W**T - W*V**T * CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), $ LDA, WORK, LDWORK, ONE, A, LDA ) @@ -266,7 +266,7 @@ $ TAU( I ), WORK, LDWORK ) * * Update the unreduced submatrix A(i+ib:n,i+ib:n), using -* an update of the form: A := A - V*W' - W*V**T +* an update of the form: A := A - V*W**T - W*V**T * CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, diff --git a/SRC/dsytri2x.f b/SRC/dsytri2x.f index 52d99532..9ab1a64c 100644 --- a/SRC/dsytri2x.f +++ b/SRC/dsytri2x.f @@ -153,7 +153,7 @@ IF( UPPER ) THEN * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL DTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -266,7 +266,7 @@ END IF END DO * -* U11T*invD1*U11->U11 +* U11**T*invD1*U11->U11 * CALL DTRMM('L','U','T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -277,13 +277,13 @@ END DO END DO * -* U01'invD*U01->A(CUT+I,CUT+J) +* U01**T*invD*U01->A(CUT+I,CUT+J) * CALL DGEMM('T','N',NNB,NNB,CUT,ONE,A(1,CUT+1),LDA, $ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* U11 = U11T*invD1*U11 + U01'invD*U01 +* U11 = U11**T*invD1*U11 + U01**T*invD*U01 * DO I=1,NNB DO J=I,NNB @@ -291,7 +291,7 @@ END DO END DO * -* U01 = U00T*invD0*U01 +* U01 = U00**T*invD0*U01 * CALL DTRMM('L',UPLO,'T','U',CUT, NNB, $ ONE,A,LDA,WORK,N+NB+1) @@ -309,7 +309,7 @@ * END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=1 DO WHILE ( I .LE. N ) @@ -331,7 +331,7 @@ * * LOWER... * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL DTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -438,7 +438,7 @@ END IF END DO * -* L11T*invD1*L11->L11 +* L11**T*invD1*L11->L11 * CALL DTRMM('L',UPLO,'T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -452,13 +452,13 @@ * IF ( (CUT+NNB) .LT. N ) THEN * -* L21T*invD2*L21->A(CUT+I,CUT+J) +* L21**T*invD2*L21->A(CUT+I,CUT+J) * CALL DGEMM('T','N',NNB,NNB,N-NNB-CUT,ONE,A(CUT+NNB+1,CUT+1) $ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* L11 = L11T*invD1*L11 + U01'invD*U01 +* L11 = L11**T*invD1*L11 + U01**T*invD*U01 * DO I=1,NNB DO J=1,I @@ -466,7 +466,7 @@ END DO END DO * -* L01 = L22T*invD2*L21 +* L01 = L22**T*invD2*L21 * CALL DTRMM('L',UPLO,'T','U', N-NNB-CUT, NNB, $ ONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1) @@ -481,7 +481,7 @@ ELSE * -* L11 = L11T*invD1*L11 +* L11 = L11**T*invD1*L11 * DO I=1,NNB DO J=1,I @@ -495,7 +495,7 @@ CUT=CUT+NNB END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=N DO WHILE ( I .GE. 1 ) diff --git a/SRC/dtbcon.f b/SRC/dtbcon.f index dfa3cb7a..88237cea 100644 --- a/SRC/dtbcon.f +++ b/SRC/dtbcon.f @@ -170,7 +170,7 @@ $ AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**T). * CALL DLATBS( UPLO, 'Transpose', DIAG, NORMIN, N, KD, AB, $ LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO ) diff --git a/SRC/dtbrfs.f b/SRC/dtbrfs.f index 2c56acd0..a1c2e7b5 100644 --- a/SRC/dtbrfs.f +++ b/SRC/dtbrfs.f @@ -195,7 +195,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), @@ -258,7 +258,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN IF( NOUNIT ) THEN diff --git a/SRC/dtbtrs.f b/SRC/dtbtrs.f index 78f312e6..d79a193e 100644 --- a/SRC/dtbtrs.f +++ b/SRC/dtbtrs.f @@ -150,7 +150,7 @@ END IF INFO = 0 * -* Solve A * X = B or A' * X = B. +* Solve A * X = B or A**T * X = B. * DO 30 J = 1, NRHS CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, B( 1, J ), 1 ) diff --git a/SRC/dtfsm.f b/SRC/dtfsm.f index 52088b56..94673aec 100644 --- a/SRC/dtfsm.f +++ b/SRC/dtfsm.f @@ -31,7 +31,7 @@ * where alpha is a scalar, X and B are m by n matrices, A is a unit, or * non-unit, upper or lower triangular matrix and op( A ) is one of * -* op( A ) = A or op( A ) = A'. +* op( A ) = A or op( A ) = A**T. * * A is in Rectangular Full Packed (RFP) Format. * diff --git a/SRC/dtgexc.f b/SRC/dtgexc.f index 56158d45..f7e2cc17 100644 --- a/SRC/dtgexc.f +++ b/SRC/dtgexc.f @@ -21,7 +21,7 @@ * DTGEXC reorders the generalized real Schur decomposition of a real * matrix pair (A,B) using an orthogonal equivalence transformation * -* (A, B) = Q * (A, B) * Z', +* (A, B) = Q * (A, B) * Z**T, * * so that the diagonal block of (A, B) with row index IFST is moved * to row ILST. diff --git a/SRC/dtgsen.f b/SRC/dtgsen.f index 583d087f..1735c006 100644 --- a/SRC/dtgsen.f +++ b/SRC/dtgsen.f @@ -212,7 +212,7 @@ * ( 0 A22),( 0 B22) n2 * n1 n2 n1 n2 * -* where N = n1+n2 and U' means the transpose of U. The first n1 columns +* where N = n1+n2 and U**T means the transpose of U. The first n1 columns * of U and W span the specified pair of left and right eigenspaces * (deflating subspaces) of (A, B). * diff --git a/SRC/dtgsja.f b/SRC/dtgsja.f index 1cb3b244..702ef939 100644 --- a/SRC/dtgsja.f +++ b/SRC/dtgsja.f @@ -249,7 +249,7 @@ * * U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1, * -* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose +* where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose * of Z. C1 and S1 are diagonal matrices satisfying * * C1**2 + S1**2 = I, diff --git a/SRC/dtgsna.f b/SRC/dtgsna.f index 4f3d1dc7..6cbc0c72 100644 --- a/SRC/dtgsna.f +++ b/SRC/dtgsna.f @@ -25,7 +25,7 @@ * eigenvalues and/or eigenvectors of a matrix pair (A, B) in * generalized real Schur canonical form (or of any matrix pair * (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where -* Z' denotes the transpose of Z. +* Z**T denotes the transpose of Z. * * (A, B) must be in generalized real Schur form (as returned by DGGES), * i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal @@ -233,8 +233,8 @@ * and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an * upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) * -* Z2 = [ kron(S11', In-2) -kron(I2, S22) ] -* [ kron(T11', In-2) -kron(I2, T22) ] +* Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ] +* [ kron(T11**T, In-2) -kron(I2, T22) ] * * Note that if the default method for computing DIF is wanted (see * DLATDF), then the parameter DIFDRI (see below) should be changed diff --git a/SRC/dtgsy2.f b/SRC/dtgsy2.f index 72dcf698..bfaa9ae5 100644 --- a/SRC/dtgsy2.f +++ b/SRC/dtgsy2.f @@ -38,19 +38,19 @@ * In matrix notation solving equation (1) corresponds to solve * Z*x = scale*b, where Z is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ], +* Z = [ kron(In, A) -kron(B**T, Im) ] (2) +* [ kron(In, D) -kron(E**T, Im) ], * -* Ik is the identity matrix of size k and X' is the transpose of X. +* Ik is the identity matrix of size k and X**T is the transpose of X. * kron(X, Y) is the Kronecker product between the matrices X and Y. * In the process of solving (1), we solve a number of such systems * where Dim(In), Dim(In) = 1 or 2. * -* If TRANS = 'T', solve the transposed system Z'*y = scale*b for y, +* If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y, * which is equivalent to solve for R and L in * -* A' * R + D' * L = scale * C (3) -* R * B' + L * E' = scale * -F +* A**T * R + D**T * L = scale * C (3) +* R * B**T + L * E**T = scale * -F * * This case is used to compute an estimate of Dif[(A, D), (B, E)] = * sigma_min(Z) using reverse communicaton with DLACON. @@ -649,7 +649,7 @@ ZDIM = MB*NB*2 IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN * -* Build a 2-by-2 system Z' * x = RHS +* Build a 2-by-2 system Z**T * x = RHS * Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = -B( JS, JS ) @@ -661,7 +661,7 @@ RHS( 1 ) = C( IS, JS ) RHS( 2 ) = F( IS, JS ) * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) @@ -703,7 +703,7 @@ * ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN * -* Build a 4-by-4 system Z' * x = RHS +* Build a 4-by-4 system Z**T * x = RHS * Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = ZERO @@ -732,7 +732,7 @@ RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( IS, JSP1 ) * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) @@ -775,7 +775,7 @@ * ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN * -* Build a 4-by-4 system Z' * x = RHS +* Build a 4-by-4 system Z**T * x = RHS * Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = A( IS, ISP1 ) @@ -804,7 +804,7 @@ RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( ISP1, JS ) * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) @@ -846,7 +846,7 @@ * ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN * -* Build an 8-by-8 system Z' * x = RHS +* Build an 8-by-8 system Z**T * x = RHS * CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ ) * @@ -898,7 +898,7 @@ 160 CONTINUE * * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) diff --git a/SRC/dtgsyl.f b/SRC/dtgsyl.f index 1b82a5c9..4fc82c87 100644 --- a/SRC/dtgsyl.f +++ b/SRC/dtgsyl.f @@ -40,10 +40,10 @@ * In matrix notation (1) is equivalent to solve Zx = scale b, where * Z is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ]. +* Z = [ kron(In, A) -kron(B**T, Im) ] (2) +* [ kron(In, D) -kron(E**T, Im) ]. * -* Here Ik is the identity matrix of size k and X' is the transpose of +* Here Ik is the identity matrix of size k and X**T is the transpose of * X. kron(X, Y) is the Kronecker product between the matrices X and Y. * * If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b, diff --git a/SRC/dtpcon.f b/SRC/dtpcon.f index f20d1839..8405dfaa 100644 --- a/SRC/dtpcon.f +++ b/SRC/dtpcon.f @@ -159,7 +159,7 @@ $ WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**T). * CALL DLATPS( UPLO, 'Transpose', DIAG, NORMIN, N, AP, $ WORK, SCALE, WORK( 2*N+1 ), INFO ) diff --git a/SRC/dtprfs.f b/SRC/dtprfs.f index f0ac8d38..ff275c14 100644 --- a/SRC/dtprfs.f +++ b/SRC/dtprfs.f @@ -184,7 +184,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) @@ -250,7 +250,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN KC = 1 diff --git a/SRC/dtptrs.f b/SRC/dtptrs.f index a5575dc5..2a350671 100644 --- a/SRC/dtptrs.f +++ b/SRC/dtptrs.f @@ -141,7 +141,7 @@ END IF INFO = 0 * -* Solve A * x = b or A' * x = b. +* Solve A * x = b or A**T * x = b. * DO 30 J = 1, NRHS CALL DTPSV( UPLO, TRANS, DIAG, N, AP, B( 1, J ), 1 ) diff --git a/SRC/dtrcon.f b/SRC/dtrcon.f index 7bb35b6e..bb522de3 100644 --- a/SRC/dtrcon.f +++ b/SRC/dtrcon.f @@ -165,7 +165,7 @@ $ LDA, WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**T). * CALL DLATRS( UPLO, 'Transpose', DIAG, NORMIN, N, A, LDA, $ WORK, SCALE, WORK( 2*N+1 ), INFO ) diff --git a/SRC/dtrevc.f b/SRC/dtrevc.f index a37e2264..0ac5725d 100644 --- a/SRC/dtrevc.f +++ b/SRC/dtrevc.f @@ -726,8 +726,8 @@ $ WORK( KI+1+N ), 1 ) * * Solve -* [T(J,J)-WR T(J,J+1) ]'* X = SCALE*( WORK1 ) -* [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 ) +* [T(J,J)-WR T(J,J+1) ]**T * X = SCALE*( WORK1 ) +* [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 ) * CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ), $ LDT, ONE, ONE, WORK( J+N ), N, WR, @@ -891,8 +891,8 @@ $ WORK( KI+2+N2 ), 1 ) * * Solve 2-by-2 complex linear equation -* ([T(j,j) T(j,j+1) ]'-(wr-i*wi)*I)*X = SCALE*B -* ([T(j+1,j) T(j+1,j+1)] ) +* ([T(j,j) T(j,j+1) ]**T-(wr-i*wi)*I)*X = SCALE*B +* ([T(j+1,j) T(j+1,j+1)] ) * CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ), $ LDT, ONE, ONE, WORK( J+N ), N, WR, diff --git a/SRC/dtrrfs.f b/SRC/dtrrfs.f index 49fedcc8..6a5c23d7 100644 --- a/SRC/dtrrfs.f +++ b/SRC/dtrrfs.f @@ -190,7 +190,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 ) @@ -250,7 +250,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN IF( NOUNIT ) THEN diff --git a/SRC/dtrsen.f b/SRC/dtrsen.f index ba5528ce..e92b2962 100644 --- a/SRC/dtrsen.f +++ b/SRC/dtrsen.f @@ -160,7 +160,7 @@ * ( 0 T22 ) n2 * n1 n2 * -* where N = n1+n2 and Z' means the transpose of Z. The first n1 columns +* where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns * of Z span the specified invariant subspace of T. * * If T has been obtained from the real Schur factorization of a matrix diff --git a/SRC/dtrtrs.f b/SRC/dtrtrs.f index 448e27fc..c83d9c64 100644 --- a/SRC/dtrtrs.f +++ b/SRC/dtrtrs.f @@ -136,7 +136,7 @@ END IF INFO = 0 * -* Solve A * x = b or A' * x = b. +* Solve A * x = b or A**T * x = b. * CALL DTRSM( 'Left', UPLO, TRANS, DIAG, N, NRHS, ONE, A, LDA, B, $ LDB ) diff --git a/SRC/sbdsdc.f b/SRC/sbdsdc.f index 95ad7cf3..883d90a2 100644 --- a/SRC/sbdsdc.f +++ b/SRC/sbdsdc.f @@ -77,7 +77,7 @@ * * VT (output) REAL array, dimension (LDVT,N) * If COMPQ = 'I', then: -* On exit, if INFO = 0, VT' contains the right singular +* On exit, if INFO = 0, VT**T contains the right singular * vectors of the bidiagonal matrix. * For other values of COMPQ, VT is not referenced. * diff --git a/SRC/sgbtrs.f b/SRC/sgbtrs.f index 8e3ecee3..ef988340 100644 --- a/SRC/sgbtrs.f +++ b/SRC/sgbtrs.f @@ -19,7 +19,7 @@ * ======= * * SGBTRS solves a system of linear equations -* A * X = B or A' * X = B +* A * X = B or A**T * X = B * with a general band matrix A using the LU factorization computed * by SGBTRF. * @@ -29,8 +29,8 @@ * TRANS (input) CHARACTER*1 * Specifies the form of the system of equations. * = 'N': A * X = B (No transpose) -* = 'T': A'* X = B (Transpose) -* = 'C': A'* X = B (Conjugate transpose = Transpose) +* = 'T': A**T* X = B (Transpose) +* = 'C': A**T* X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The order of the matrix A. N >= 0. @@ -157,17 +157,17 @@ * ELSE * -* Solve A'*X = B. +* Solve A**T*X = B. * DO 30 I = 1, NRHS * -* Solve U'*X = B, overwriting B with X. +* Solve U**T*X = B, overwriting B with X. * CALL STBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU, AB, $ LDAB, B( 1, I ), 1 ) 30 CONTINUE * -* Solve L'*X = B, overwriting B with X. +* Solve L**T*X = B, overwriting B with X. * IF( LNOTI ) THEN DO 40 J = N - 1, 1, -1 diff --git a/SRC/sgebd2.f b/SRC/sgebd2.f index 4a782f72..fa0bc55c 100644 --- a/SRC/sgebd2.f +++ b/SRC/sgebd2.f @@ -17,7 +17,7 @@ * ======= * * SGEBD2 reduces a real general m by n matrix A to upper or lower -* bidiagonal form B by an orthogonal transformation: Q' * A * P = B. +* bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. * * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. * diff --git a/SRC/sgels.f b/SRC/sgels.f index d20e4f02..6fc82655 100644 --- a/SRC/sgels.f +++ b/SRC/sgels.f @@ -298,9 +298,9 @@ * ELSE * -* Overdetermined system of equations A' * X = B +* Overdetermined system of equations A**T * X = B * -* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) +* B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS) * CALL STRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS, $ A, LDA, B, LDB, INFO ) @@ -371,7 +371,7 @@ * ELSE * -* overdetermined system min || A' * X - B || +* overdetermined system min || A**T * X - B || * * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) * @@ -381,7 +381,7 @@ * * workspace at least NRHS, optimally NRHS*NB * -* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) +* B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS) * CALL STRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS, $ A, LDA, B, LDB, INFO ) diff --git a/SRC/sggsvd.f b/SRC/sggsvd.f index 61405d71..322d478b 100644 --- a/SRC/sggsvd.f +++ b/SRC/sggsvd.f @@ -128,7 +128,7 @@ * L (output) INTEGER * On exit, K and L specify the dimension of the subblocks * described in the Purpose section. -* K + L = effective numerical rank of (A',B')**T. +* K + L = effective numerical rank of (A**T,B**T)**T. * * A (input/output) REAL array, dimension (LDA,N) * On entry, the M-by-N matrix A. @@ -209,7 +209,7 @@ * TOLA REAL * TOLB REAL * TOLA and TOLB are the thresholds to determine the effective -* rank of (A',B')**T. Generally, they are set to +* rank of (A**T,B**T)**T. Generally, they are set to * TOLA = MAX(M,N)*norm(A)*MACHEPS, * TOLB = MAX(P,N)*norm(B)*MACHEPS. * The size of TOLA and TOLB may affect the size of backward diff --git a/SRC/sgttrs.f b/SRC/sgttrs.f index 239aa01a..9e05e924 100644 --- a/SRC/sgttrs.f +++ b/SRC/sgttrs.f @@ -19,7 +19,7 @@ * ======= * * SGTTRS solves one of the systems of equations -* A*X = B or A'*X = B, +* A*X = B or A**T*X = B, * with a tridiagonal matrix A using the LU factorization computed * by SGTTRF. * @@ -29,8 +29,8 @@ * TRANS (input) CHARACTER*1 * Specifies the form of the system of equations. * = 'N': A * X = B (No transpose) -* = 'T': A'* X = B (Transpose) -* = 'C': A'* X = B (Conjugate transpose = Transpose) +* = 'T': A**T* X = B (Transpose) +* = 'C': A**T* X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The order of the matrix A. diff --git a/SRC/sgtts2.f b/SRC/sgtts2.f index 67245e07..293d3f04 100644 --- a/SRC/sgtts2.f +++ b/SRC/sgtts2.f @@ -17,7 +17,7 @@ * ======= * * SGTTS2 solves one of the systems of equations -* A*X = B or A'*X = B, +* A*X = B or A**T*X = B, * with a tridiagonal matrix A using the LU factorization computed * by SGTTRF. * @@ -27,8 +27,8 @@ * ITRANS (input) INTEGER * Specifies the form of the system of equations. * = 0: A * X = B (No transpose) -* = 1: A'* X = B (Transpose) -* = 2: A'* X = B (Conjugate transpose = Transpose) +* = 1: A**T* X = B (Transpose) +* = 2: A**T* X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The order of the matrix A. @@ -138,11 +138,11 @@ END IF ELSE * -* Solve A' * X = B. +* Solve A**T * X = B. * IF( NRHS.LE.1 ) THEN * -* Solve U'*x = b. +* Solve U**T*x = b. * J = 1 70 CONTINUE @@ -154,7 +154,7 @@ $ B( I-2, J ) ) / D( I ) 80 CONTINUE * -* Solve L'*x = b. +* Solve L**T*x = b. * DO 90 I = N - 1, 1, -1 IP = IPIV( I ) @@ -170,7 +170,7 @@ ELSE DO 120 J = 1, NRHS * -* Solve U'*x = b. +* Solve U**T*x = b. * B( 1, J ) = B( 1, J ) / D( 1 ) IF( N.GT.1 ) diff --git a/SRC/sla_gbrcond.f b/SRC/sla_gbrcond.f index bc1e494d..936cb5cc 100644 --- a/SRC/sla_gbrcond.f +++ b/SRC/sla_gbrcond.f @@ -225,7 +225,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/sla_gercond.f b/SRC/sla_gercond.f index ab1e7f3c..acf8fbcc 100644 --- a/SRC/sla_gercond.f +++ b/SRC/sla_gercond.f @@ -207,7 +207,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/sla_porcond.f b/SRC/sla_porcond.f index ad538253..8b9b1bf5 100644 --- a/SRC/sla_porcond.f +++ b/SRC/sla_porcond.f @@ -212,7 +212,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/sla_syrcond.f b/SRC/sla_syrcond.f index 0b3833be..2f941889 100644 --- a/SRC/sla_syrcond.f +++ b/SRC/sla_syrcond.f @@ -218,7 +218,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CMODE .EQ. 1 ) THEN DO I = 1, N diff --git a/SRC/slacn2.f b/SRC/slacn2.f index 6c47c1ed..db8109e8 100644 --- a/SRC/slacn2.f +++ b/SRC/slacn2.f @@ -33,7 +33,7 @@ * X (input/output) REAL array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, +* A**T * X, if KASE=2, * and SLACN2 must be re-called with all the other parameters * unchanged. * @@ -47,7 +47,7 @@ * KASE (input/output) INTEGER * On the initial call to SLACN2, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**T * X. * On the final return from SLACN2, KASE will again be 0. * * ISAVE (input/output) INTEGER array, dimension (3) diff --git a/SRC/slacon.f b/SRC/slacon.f index 0d149b07..9f2fd02c 100644 --- a/SRC/slacon.f +++ b/SRC/slacon.f @@ -33,7 +33,7 @@ * X (input/output) REAL array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, +* A**T * X, if KASE=2, * and SLACON must be re-called with all the other parameters * unchanged. * @@ -47,7 +47,7 @@ * KASE (input/output) INTEGER * On the initial call to SLACON, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**T * X. * On the final return from SLACON, KASE will again be 0. * * Further Details diff --git a/SRC/slaed1.f b/SRC/slaed1.f index 58758d36..971c0bf3 100644 --- a/SRC/slaed1.f +++ b/SRC/slaed1.f @@ -26,9 +26,9 @@ * of a full symmetric matrix (which was reduced to tridiagonal form) * are desired. * -* T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out) +* T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) * -* where Z = Q'u, u is a vector of length N with ones in the +* where Z = Q**T*u, u is a vector of length N with ones in the * CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. * * The eigenvectors of the original matrix are stored in Q, and the diff --git a/SRC/slaein.f b/SRC/slaein.f index 557d20cb..d868de9a 100644 --- a/SRC/slaein.f +++ b/SRC/slaein.f @@ -427,7 +427,7 @@ VCRIT = BIGNUM * * Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector, -* or U'*(xr,xi) = scale*(vr,vi) for a left eigenvector, +* or U**T*(xr,xi) = scale*(vr,vi) for a left eigenvector, * overwriting (xr,xi) on (vr,vi). * DO 250 I = I1, I2, I3 diff --git a/SRC/slags2.f b/SRC/slags2.f index e2b854c3..36c8dedb 100644 --- a/SRC/slags2.f +++ b/SRC/slags2.f @@ -37,7 +37,7 @@ * U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) * ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) * -* Z' denotes the transpose of Z. +* Z**T denotes the transpose of Z. * * * Arguments diff --git a/SRC/slagv2.f b/SRC/slagv2.f index 53ebee30..666786ed 100644 --- a/SRC/slagv2.f +++ b/SRC/slagv2.f @@ -242,7 +242,7 @@ CALL SLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR, $ CSR, SNL, CSL ) * -* Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and +* Form (A,B) := Q(A,B)Z**T where Q is left rotation matrix and * Z is right rotation matrix computed from SLASV2 * CALL SROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) diff --git a/SRC/slahr2.f b/SRC/slahr2.f index 1d63053a..1aaf7b78 100644 --- a/SRC/slahr2.f +++ b/SRC/slahr2.f @@ -149,7 +149,7 @@ CALL SGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY, $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 ) * -* Apply I - V * T' * V**T to this column (call it b) from the +* Apply I - V * T**T * V**T to this column (call it b) from the * left, using the last column of T as workspace * * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) @@ -157,7 +157,7 @@ * * where V1 is unit lower triangular * -* w := V1' * b1 +* w := V1**T * b1 * CALL SCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL STRMV( 'Lower', 'Transpose', 'UNIT', diff --git a/SRC/slaic1.f b/SRC/slaic1.f index ab4680a8..385bcdee 100644 --- a/SRC/slaic1.f +++ b/SRC/slaic1.f @@ -27,15 +27,15 @@ * [ s*x ] * xhat = [ c ] * is an approximate singular vector of -* [ L 0 ] -* Lhat = [ w' gamma ] +* [ L 0 ] +* Lhat = [ w**T gamma ] * in the sense that * twonorm(Lhat*xhat) = sestpr. * * Depending on JOB, an estimate for the largest or smallest singular * value is computed. * -* Note that [s c]' and sestpr**2 is an eigenpair of the system +* Note that [s c]**T and sestpr**2 is an eigenpair of the system * * diag(sest*sest, 0) + [alpha gamma] * [ alpha ] * [ gamma ] diff --git a/SRC/slaln2.f b/SRC/slaln2.f index b3417441..21afed48 100644 --- a/SRC/slaln2.f +++ b/SRC/slaln2.f @@ -19,8 +19,8 @@ * ======= * * SLALN2 solves a system of the form (ca A - w D ) X = s B -* or (ca A' - w D) X = s B with possible scaling ("s") and -* perturbation of A. (A' means A-transpose.) +* or (ca A**T - w D) X = s B with possible scaling ("s") and +* perturbation of A. (A**T means A-transpose.) * * A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA * real diagonal matrix, w is a real or complex value, and X and B are @@ -257,7 +257,7 @@ * * 2x2 System * -* Compute the real part of C = ca A - w D (or ca A' - w D ) +* Compute the real part of C = ca A - w D (or ca A**T - w D ) * CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1 CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2 diff --git a/SRC/slalsa.f b/SRC/slalsa.f index f94c60ec..dd8d5e92 100644 --- a/SRC/slalsa.f +++ b/SRC/slalsa.f @@ -81,7 +81,7 @@ * POLES, GIVNUM, and Z. * * VT (input) REAL array, dimension ( LDU, SMLSIZ+1 ). -* On entry, VT' contains the right singular vector matrices of +* On entry, VT**T contains the right singular vector matrices of * all subproblems at the bottom level. * * K (input) INTEGER array, dimension ( N ). diff --git a/SRC/slansf.f b/SRC/slansf.f index 8ec70f3c..d960ef22 100644 --- a/SRC/slansf.f +++ b/SRC/slansf.f @@ -725,7 +725,7 @@ ELSE * A is xpose IF( ILU.EQ.0 ) THEN -* A' is upper +* A**T is upper DO J = 1, K - 2 CALL SLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE, S ) * U at A(0,k) @@ -746,7 +746,7 @@ CALL SLASSQ( K, A( 0+( K-1 )*LDA ), LDA+1, SCALE, S ) * tri L at A(0,k-1) ELSE -* A' is lower +* A**T is lower DO J = 1, K - 1 CALL SLASSQ( J, A( 0+J*LDA ), 1, SCALE, S ) * U at A(0,0) @@ -807,7 +807,7 @@ ELSE * A is xpose IF( ILU.EQ.0 ) THEN -* A' is upper +* A**T is upper DO J = 1, K - 1 CALL SLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE, S ) * U at A(0,k+1) @@ -828,7 +828,7 @@ CALL SLASSQ( K, A( 0+K*LDA ), LDA+1, SCALE, S ) * tri L at A(0,k) ELSE -* A' is lower +* A**T is lower DO J = 1, K - 1 CALL SLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE, S ) * U at A(0,1) diff --git a/SRC/slaqps.f b/SRC/slaqps.f index ed99d005..f9c65c63 100644 --- a/SRC/slaqps.f +++ b/SRC/slaqps.f @@ -76,7 +76,7 @@ * Auxiliar vector. * * F (input/output) REAL array, dimension (LDF,NB) -* Matrix F' = L*Y'*A. +* Matrix F**T = L*Y**T*A. * * LDF (input) INTEGER * The leading dimension of the array F. LDF >= max(1,N). diff --git a/SRC/slaqr5.f b/SRC/slaqr5.f index 6439ad76..649527a7 100644 --- a/SRC/slaqr5.f +++ b/SRC/slaqr5.f @@ -642,14 +642,14 @@ CALL SLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ), $ LDH, WH( KZS+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**T ==== * CALL SLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH ) CALL STRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE, $ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ), $ LDWH ) * -* ==== Multiply top of H by U11' ==== +* ==== Multiply top of H by U11**T ==== * CALL SGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU, $ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH ) @@ -659,7 +659,7 @@ CALL SLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH, $ WH( I2+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**T ==== * CALL STRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE, $ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH ) diff --git a/SRC/slaqtr.f b/SRC/slaqtr.f index 46843f02..e87c21be 100644 --- a/SRC/slaqtr.f +++ b/SRC/slaqtr.f @@ -36,7 +36,7 @@ * [ . ] * [ w ] * -* op(A) = A or A', A' denotes the conjugate transpose of +* op(A) = A or A**T, A**T denotes the transpose of * matrix A. * * On input, X = [ c ]. On output, X = [ p ]. @@ -290,7 +290,7 @@ * ELSE * -* Solve T'*p = scale*c +* Solve T**T*p = scale*c * JNEXT = 1 DO 40 J = 1, N diff --git a/SRC/slarfb.f b/SRC/slarfb.f index 597aaafe..40011874 100644 --- a/SRC/slarfb.f +++ b/SRC/slarfb.f @@ -286,7 +286,7 @@ LASTV = MAX( K, ILASLR( M, K, V, LDV ) ) LASTC = ILASLC( LASTV, N, C, LDC ) * -* W := C' * V = (C1**T * V1 + C2**T * V2) (stored in WORK) +* W := C**T * V = (C1**T * V1 + C2**T * V2) (stored in WORK) * * W := C2**T * @@ -373,7 +373,7 @@ CALL STRMM( 'Right', 'Lower', TRANS, 'Non-unit', $ LASTC, K, ONE, T, LDT, WORK, LDWORK ) * -* C := C - W * V' +* C := C - W * V**T * IF( LASTV.GT.K ) THEN * diff --git a/SRC/slartgs.f b/SRC/slartgs.f index a9493500..d5fb7881 100644 --- a/SRC/slartgs.f +++ b/SRC/slartgs.f @@ -61,7 +61,7 @@ * THRESH = SLAMCH('E') * -* Compute the first column of B'*B - SIGMA^2*I, up to a scale +* Compute the first column of B**T*B - SIGMA^2*I, up to a scale * factor. * IF( (SIGMA .EQ. ZERO .AND. ABS(X) .LT. THRESH) .OR. diff --git a/SRC/slarzb.f b/SRC/slarzb.f index 1c335cc8..a0b820fc 100644 --- a/SRC/slarzb.f +++ b/SRC/slarzb.f @@ -155,7 +155,7 @@ $ CALL SGEMM( 'Transpose', 'Transpose', N, K, L, ONE, $ C( M-L+1, 1 ), LDC, V, LDV, ONE, WORK, LDWORK ) * -* W( 1:n, 1:k ) = W( 1:n, 1:k ) * T' or W( 1:m, 1:k ) * T +* W( 1:n, 1:k ) = W( 1:n, 1:k ) * T**T or W( 1:m, 1:k ) * T * CALL STRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, ONE, T, $ LDT, WORK, LDWORK ) diff --git a/SRC/slarzt.f b/SRC/slarzt.f index 077ee709..b8689fa5 100644 --- a/SRC/slarzt.f +++ b/SRC/slarzt.f @@ -27,12 +27,12 @@ * If STOREV = 'C', the vector which defines the elementary reflector * H(i) is stored in the i-th column of the array V, and * -* H = I - V * T * V' +* H = I - V * T * V**T * * If STOREV = 'R', the vector which defines the elementary reflector * H(i) is stored in the i-th row of the array V, and * -* H = I - V' * T * V +* H = I - V**T * T * V * * Currently, only STOREV = 'R' and DIRECT = 'B' are supported. * diff --git a/SRC/slasd0.f b/SRC/slasd0.f index 87c8d2e6..af229b3b 100644 --- a/SRC/slasd0.f +++ b/SRC/slasd0.f @@ -55,7 +55,7 @@ * On entry, leading dimension of U. * * VT (output) REAL array, dimension at least (LDVT, M) -* On exit, VT' contains the right singular vectors. +* On exit, VT**T contains the right singular vectors. * * LDVT (input) INTEGER * On entry, leading dimension of VT. diff --git a/SRC/slasd1.f b/SRC/slasd1.f index 4fae39ae..080a9f28 100644 --- a/SRC/slasd1.f +++ b/SRC/slasd1.f @@ -26,13 +26,13 @@ * * SLASD1 computes the SVD as follows: * -* ( D1(in) 0 0 0 ) -* B = U(in) * ( Z1' a Z2' b ) * VT(in) -* ( 0 0 D2(in) 0 ) +* ( D1(in) 0 0 0 ) +* B = U(in) * ( Z1**T a Z2**T b ) * VT(in) +* ( 0 0 D2(in) 0 ) * * = U(out) * ( D(out) 0) * VT(out) * -* where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M +* where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M * with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros * elsewhere; and the entry b is empty if SQRE = 0. * @@ -101,7 +101,7 @@ * On entry VT(1:NL+1, 1:NL+1)**T contains the right singular * vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains * the right singular vectors of the lower block. On exit -* VT' contains the right singular vectors of the +* VT**T contains the right singular vectors of the * bidiagonal matrix. * * LDVT (input) INTEGER diff --git a/SRC/slasd2.f b/SRC/slasd2.f index 64c84c6a..0c4271e9 100644 --- a/SRC/slasd2.f +++ b/SRC/slasd2.f @@ -79,10 +79,10 @@ * The leading dimension of the array U. LDU >= N. * * VT (input/output) REAL array, dimension (LDVT,M) -* On entry VT' contains the right singular vectors of two +* On entry VT**T contains the right singular vectors of two * submatrices in the two square blocks with corners at (1,1), * (NL+1, NL+1), and (NL+2, NL+2), (M,M). -* On exit VT' contains the trailing (N-K) updated right singular +* On exit VT**T contains the trailing (N-K) updated right singular * vectors (those which were deflated) in its last N-K columns. * In case SQRE =1, the last row of VT spans the right null * space. @@ -107,7 +107,7 @@ * The leading dimension of the array U2. LDU2 >= N. * * VT2 (output) REAL array, dimension (LDVT2,N) -* VT2' contains a copy of the first K right singular vectors +* VT2**T contains a copy of the first K right singular vectors * which will be used by SLASD3 in a matrix multiply (SGEMM) to * solve for the new right singular vectors. VT2 is arranged into * three blocks. The first block contains a row that corresponds diff --git a/SRC/slasd3.f b/SRC/slasd3.f index 37ad873f..62f0a232 100644 --- a/SRC/slasd3.f +++ b/SRC/slasd3.f @@ -84,14 +84,14 @@ * The leading dimension of the array U2. LDU2 >= N. * * VT (output) REAL array, dimension (LDVT, M) -* The last M - K columns of VT' contain the deflated +* The last M - K columns of VT**T contain the deflated * right singular vectors. * * LDVT (input) INTEGER * The leading dimension of the array VT. LDVT >= N. * * VT2 (input/output) REAL array, dimension (LDVT2, N) -* The first K columns of VT2' contain the non-deflated +* The first K columns of VT2**T contain the non-deflated * right singular vectors for the split problem. * * LDVT2 (input) INTEGER diff --git a/SRC/slasd6.f b/SRC/slasd6.f index 7b996587..9bf0f1c8 100644 --- a/SRC/slasd6.f +++ b/SRC/slasd6.f @@ -34,13 +34,13 @@ * * SLASD6 computes the SVD as follows: * -* ( D1(in) 0 0 0 ) -* B = U(in) * ( Z1' a Z2' b ) * VT(in) -* ( 0 0 D2(in) 0 ) +* ( D1(in) 0 0 0 ) +* B = U(in) * ( Z1**T a Z2**T b ) * VT(in) +* ( 0 0 D2(in) 0 ) * * = U(out) * ( D(out) 0) * VT(out) * -* where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M +* where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M * with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros * elsewhere; and the entry b is empty if SQRE = 0. * diff --git a/SRC/slasda.f b/SRC/slasda.f index cb8499e3..4a501c67 100644 --- a/SRC/slasda.f +++ b/SRC/slasda.f @@ -75,7 +75,7 @@ * * VT (output) REAL array, * dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced -* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right +* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right * singular vector matrices of all subproblems at the bottom * level. * diff --git a/SRC/slasdq.f b/SRC/slasdq.f index 84f74fe7..eaf5296c 100644 --- a/SRC/slasdq.f +++ b/SRC/slasdq.f @@ -22,12 +22,12 @@ * (upper or lower) bidiagonal matrix with diagonal D and offdiagonal * E, accumulating the transformations if desired. Letting B denote * the input bidiagonal matrix, the algorithm computes orthogonal -* matrices Q and P such that B = Q * S * P' (P' denotes the transpose +* matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose * of P). The singular values S are overwritten on D. * * The input matrix U is changed to U * Q if desired. -* The input matrix VT is changed to P' * VT if desired. -* The input matrix C is changed to Q' * C if desired. +* The input matrix VT is changed to P**T * VT if desired. +* The input matrix C is changed to Q**T * C if desired. * * See "Computing Small Singular Values of Bidiagonal Matrices With * Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, @@ -84,7 +84,7 @@ * * VT (input/output) REAL array, dimension (LDVT, NCVT) * On entry, contains a matrix which on exit has been -* premultiplied by P', dimension N-by-NCVT if SQRE = 0 +* premultiplied by P**T, dimension N-by-NCVT if SQRE = 0 * and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0). * * LDVT (input) INTEGER @@ -104,7 +104,7 @@ * * C (input/output) REAL array, dimension (LDC, NCC) * On entry, contains an N-by-NCC matrix which on exit -* has been premultiplied by Q' dimension N-by-NCC if SQRE = 0 +* has been premultiplied by Q**T dimension N-by-NCC if SQRE = 0 * and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0). * * LDC (input) INTEGER diff --git a/SRC/slasr.f b/SRC/slasr.f index a419de28..89a95434 100644 --- a/SRC/slasr.f +++ b/SRC/slasr.f @@ -272,7 +272,7 @@ END IF ELSE IF( LSAME( SIDE, 'R' ) ) THEN * -* Form A * P' +* Form A * P**T * IF( LSAME( PIVOT, 'V' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN diff --git a/SRC/slasy2.f b/SRC/slasy2.f index 860966e6..f58d8d83 100644 --- a/SRC/slasy2.f +++ b/SRC/slasy2.f @@ -24,7 +24,7 @@ * op(TL)*X + ISGN*X*op(TR) = SCALE*B, * * where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or -* -1. op(T) = T or T', where T' denotes the transpose of T. +* -1. op(T) = T or T**T, where T**T denotes the transpose of T. * * Arguments * ========= @@ -32,12 +32,12 @@ * LTRANL (input) LOGICAL * On entry, LTRANL specifies the op(TL): * = .FALSE., op(TL) = TL, -* = .TRUE., op(TL) = TL'. +* = .TRUE., op(TL) = TL**T. * * LTRANR (input) LOGICAL * On entry, LTRANR specifies the op(TR): * = .FALSE., op(TR) = TR, -* = .TRUE., op(TR) = TR'. +* = .TRUE., op(TR) = TR**T. * * ISGN (input) INTEGER * On entry, ISGN specifies the sign of the equation diff --git a/SRC/slatbs.f b/SRC/slatbs.f index 0d862347..07bc2bdf 100644 --- a/SRC/slatbs.f +++ b/SRC/slatbs.f @@ -20,10 +20,10 @@ * * SLATBS solves one of the triangular systems * -* A *x = s*b or A'*x = s*b +* A *x = s*b or A**T*x = s*b * * with scaling to prevent overflow, where A is an upper or lower -* triangular band matrix. Here A' denotes the transpose of A, x and b +* triangular band matrix. Here A**T denotes the transpose of A, x and b * are n-element vectors, and s is a scaling factor, usually less than * or equal to 1, chosen so that the components of x will be less than * the overflow threshold. If the unscaled problem will not cause @@ -42,8 +42,8 @@ * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': Solve A * x = s*b (No transpose) -* = 'T': Solve A'* x = s*b (Transpose) -* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) +* = 'T': Solve A**T* x = s*b (Transpose) +* = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. @@ -79,7 +79,7 @@ * * SCALE (output) REAL * The scaling factor s for the triangular system -* A * x = s*b or A'* x = s*b. +* A * x = s*b or A**T* x = s*b. * If SCALE = 0, the matrix A is singular or badly scaled, and * the vector x is an exact or approximate solution to A*x = 0. * @@ -145,15 +145,15 @@ * prevent overflow, but if the bound overflows, x is set to 0, x(j) to * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. * -* Similarly, a row-wise scheme is used to solve A'*x = b. The basic +* Similarly, a row-wise scheme is used to solve A**T*x = b. The basic * algorithm for A upper triangular is * * for j = 1, ..., n -* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) +* x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) * end * * We simultaneously compute two bounds -* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j +* G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j * M(j) = bound on x(i), 1<=i<=j * * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we @@ -356,7 +356,7 @@ * ELSE * -* Compute the growth in A' * x = b. +* Compute the growth in A**T * x = b. * IF( UPPER ) THEN JFIRST = 1 @@ -569,7 +569,7 @@ * ELSE * -* Solve A' * x = b +* Solve A**T * x = b * DO 140 J = JFIRST, JLAST, JINC * @@ -688,7 +688,7 @@ ELSE * * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and -* scale = 0, and compute a solution to A'*x = 0. +* scale = 0, and compute a solution to A**T*x = 0. * DO 130 I = 1, N X( I ) = ZERO diff --git a/SRC/slatps.f b/SRC/slatps.f index d5f3c0f5..865ff4a1 100644 --- a/SRC/slatps.f +++ b/SRC/slatps.f @@ -20,10 +20,10 @@ * * SLATPS solves one of the triangular systems * -* A *x = s*b or A'*x = s*b +* A *x = s*b or A**T*x = s*b * * with scaling to prevent overflow, where A is an upper or lower -* triangular matrix stored in packed form. Here A' denotes the +* triangular matrix stored in packed form. Here A**T denotes the * transpose of A, x and b are n-element vectors, and s is a scaling * factor, usually less than or equal to 1, chosen so that the * components of x will be less than the overflow threshold. If the @@ -42,8 +42,8 @@ * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': Solve A * x = s*b (No transpose) -* = 'T': Solve A'* x = s*b (Transpose) -* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) +* = 'T': Solve A**T* x = s*b (Transpose) +* = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. @@ -72,7 +72,7 @@ * * SCALE (output) REAL * The scaling factor s for the triangular system -* A * x = s*b or A'* x = s*b. +* A * x = s*b or A**T* x = s*b. * If SCALE = 0, the matrix A is singular or badly scaled, and * the vector x is an exact or approximate solution to A*x = 0. * @@ -138,15 +138,15 @@ * prevent overflow, but if the bound overflows, x is set to 0, x(j) to * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. * -* Similarly, a row-wise scheme is used to solve A'*x = b. The basic +* Similarly, a row-wise scheme is used to solve A**T*x = b. The basic * algorithm for A upper triangular is * * for j = 1, ..., n -* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) +* x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) * end * * We simultaneously compute two bounds -* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j +* G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j * M(j) = bound on x(i), 1<=i<=j * * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we @@ -346,7 +346,7 @@ * ELSE * -* Compute the growth in A' * x = b. +* Compute the growth in A**T * x = b. * IF( UPPER ) THEN JFIRST = 1 @@ -561,7 +561,7 @@ * ELSE * -* Solve A' * x = b +* Solve A**T * x = b * IP = JFIRST*( JFIRST+1 ) / 2 JLEN = 1 @@ -675,7 +675,7 @@ ELSE * * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and -* scale = 0, and compute a solution to A'*x = 0. +* scale = 0, and compute a solution to A**T*x = 0. * DO 130 I = 1, N X( I ) = ZERO diff --git a/SRC/slatrs.f b/SRC/slatrs.f index ed0338b7..63e5f099 100644 --- a/SRC/slatrs.f +++ b/SRC/slatrs.f @@ -20,10 +20,10 @@ * * SLATRS solves one of the triangular systems * -* A *x = s*b or A'*x = s*b +* A *x = s*b or A**T*x = s*b * * with scaling to prevent overflow. Here A is an upper or lower -* triangular matrix, A' denotes the transpose of A, x and b are +* triangular matrix, A**T denotes the transpose of A, x and b are * n-element vectors, and s is a scaling factor, usually less than * or equal to 1, chosen so that the components of x will be less than * the overflow threshold. If the unscaled problem will not cause @@ -42,8 +42,8 @@ * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': Solve A * x = s*b (No transpose) -* = 'T': Solve A'* x = s*b (Transpose) -* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) +* = 'T': Solve A**T* x = s*b (Transpose) +* = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. @@ -78,7 +78,7 @@ * * SCALE (output) REAL * The scaling factor s for the triangular system -* A * x = s*b or A'* x = s*b. +* A * x = s*b or A**T* x = s*b. * If SCALE = 0, the matrix A is singular or badly scaled, and * the vector x is an exact or approximate solution to A*x = 0. * @@ -144,15 +144,15 @@ * prevent overflow, but if the bound overflows, x is set to 0, x(j) to * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. * -* Similarly, a row-wise scheme is used to solve A'*x = b. The basic +* Similarly, a row-wise scheme is used to solve A**T*x = b. The basic * algorithm for A upper triangular is * * for j = 1, ..., n -* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) +* x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) * end * * We simultaneously compute two bounds -* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j +* G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j * M(j) = bound on x(i), 1<=i<=j * * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we @@ -346,7 +346,7 @@ * ELSE * -* Compute the growth in A' * x = b. +* Compute the growth in A**T * x = b. * IF( UPPER ) THEN JFIRST = 1 @@ -554,7 +554,7 @@ * ELSE * -* Solve A' * x = b +* Solve A**T * x = b * DO 140 J = JFIRST, JLAST, JINC * @@ -666,7 +666,7 @@ ELSE * * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and -* scale = 0, and compute a solution to A'*x = 0. +* scale = 0, and compute a solution to A**T*x = 0. * DO 130 I = 1, N X( I ) = ZERO diff --git a/SRC/sorgbr.f b/SRC/sorgbr.f index f8460c46..9e070710 100644 --- a/SRC/sorgbr.f +++ b/SRC/sorgbr.f @@ -201,7 +201,7 @@ END IF ELSE * -* Form P', determined by a call to SGEBRD to reduce a k-by-n +* Form P**T, determined by a call to SGEBRD to reduce a k-by-n * matrix * IF( K.LT.N ) THEN @@ -215,7 +215,7 @@ * If k >= n, assume m = n * * Shift the vectors which define the elementary reflectors one -* row downward, and set the first row and column of P' to +* row downward, and set the first row and column of P**T to * those of the unit matrix * A( 1, 1 ) = ONE @@ -230,7 +230,7 @@ 60 CONTINUE IF( N.GT.1 ) THEN * -* Form P'(2:n,2:n) +* Form P**T(2:n,2:n) * CALL SORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK, $ LWORK, IINFO ) diff --git a/SRC/spbrfs.f b/SRC/spbrfs.f index b321102f..3f33b3f1 100644 --- a/SRC/spbrfs.f +++ b/SRC/spbrfs.f @@ -304,7 +304,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, $ INFO ) diff --git a/SRC/sporfs.f b/SRC/sporfs.f index 6c492348..a52ae17f 100644 --- a/SRC/sporfs.f +++ b/SRC/sporfs.f @@ -296,7 +296,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL SPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) DO 110 I = 1, N diff --git a/SRC/spprfs.f b/SRC/spprfs.f index 197595b4..ec29836f 100644 --- a/SRC/spprfs.f +++ b/SRC/spprfs.f @@ -293,7 +293,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL SPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) DO 110 I = 1, N diff --git a/SRC/spstf2.f b/SRC/spstf2.f index 8035d6ad..851e4b60 100644 --- a/SRC/spstf2.f +++ b/SRC/spstf2.f @@ -21,8 +21,8 @@ * pivoting of a real symmetric positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**T * U , if UPLO = 'U', +* P**T * A * P = L * L**T, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -161,7 +161,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**T * U * DO 130 J = 1, N * @@ -224,7 +224,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**T * DO 150 J = 1, N * diff --git a/SRC/spstrf.f b/SRC/spstrf.f index 79b157f3..0b70e609 100644 --- a/SRC/spstrf.f +++ b/SRC/spstrf.f @@ -21,8 +21,8 @@ * pivoting of a real symmetric positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**T * U , if UPLO = 'U', +* P**T * A * P = L * L**T, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -170,7 +170,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**T * U * DO 140 K = 1, N, NB * @@ -257,7 +257,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**T * DO 180 K = 1, N, NB * diff --git a/SRC/sptcon.f b/SRC/sptcon.f index 72666776..bf47778a 100644 --- a/SRC/sptcon.f +++ b/SRC/sptcon.f @@ -117,7 +117,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**T. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T. * * Solve M(L) * x = e. * diff --git a/SRC/sptrfs.f b/SRC/sptrfs.f index 2b7e3b68..875d76b2 100644 --- a/SRC/sptrfs.f +++ b/SRC/sptrfs.f @@ -263,7 +263,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**T. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T. * * Solve M(L) * x = e. * diff --git a/SRC/ssfrk.f b/SRC/ssfrk.f index 956e1755..51e2bb48 100644 --- a/SRC/ssfrk.f +++ b/SRC/ssfrk.f @@ -26,11 +26,11 @@ * * SSFRK performs one of the symmetric rank--k operations * -* C := alpha*A*A' + beta*C, +* C := alpha*A*A**T + beta*C, * * or * -* C := alpha*A'*A + beta*C, +* C := alpha*A**T*A + beta*C, * * where alpha and beta are real scalars, C is an n--by--n symmetric * matrix and A is an n--by--k matrix in the first case and a k--by--n @@ -60,9 +60,9 @@ * On entry, TRANS specifies the operation to be performed as * follows: * -* TRANS = 'N' or 'n' C := alpha*A*A' + beta*C. +* TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C. * -* TRANS = 'T' or 't' C := alpha*A'*A + beta*C. +* TRANS = 'T' or 't' C := alpha*A**T*A + beta*C. * * Unchanged on exit. * diff --git a/SRC/ssprfs.f b/SRC/ssprfs.f index 656ebfab..f6b39d49 100644 --- a/SRC/ssprfs.f +++ b/SRC/ssprfs.f @@ -298,7 +298,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL SSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, $ INFO ) diff --git a/SRC/ssptrd.f b/SRC/ssptrd.f index 4599f503..8ebd3dd1 100644 --- a/SRC/ssptrd.f +++ b/SRC/ssptrd.f @@ -207,7 +207,7 @@ CALL SAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**T +* A := A - v * w**T - w * v**T * CALL SSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, $ AP( I1I1 ) ) diff --git a/SRC/ssyrfs.f b/SRC/ssyrfs.f index 25ab28d9..a3c5f10c 100644 --- a/SRC/ssyrfs.f +++ b/SRC/ssyrfs.f @@ -302,7 +302,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL SSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, $ INFO ) diff --git a/SRC/ssytd2.f b/SRC/ssytd2.f index e15280ee..19fe1057 100644 --- a/SRC/ssytd2.f +++ b/SRC/ssytd2.f @@ -228,7 +228,7 @@ CALL SAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**T +* A := A - v * w**T - w * v**T * CALL SSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, $ A( I+1, I+1 ), LDA ) diff --git a/SRC/ssytrd.f b/SRC/ssytrd.f index b122d9c2..bb8b1647 100644 --- a/SRC/ssytrd.f +++ b/SRC/ssytrd.f @@ -235,7 +235,7 @@ $ LDWORK ) * * Update the unreduced submatrix A(1:i-1,1:i-1), using an -* update of the form: A := A - V*W' - W*V**T +* update of the form: A := A - V*W**T - W*V**T * CALL SSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), $ LDA, WORK, LDWORK, ONE, A, LDA ) @@ -266,7 +266,7 @@ $ TAU( I ), WORK, LDWORK ) * * Update the unreduced submatrix A(i+ib:n,i+ib:n), using -* an update of the form: A := A - V*W' - W*V**T +* an update of the form: A := A - V*W**T - W*V**T * CALL SSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, diff --git a/SRC/ssytri2x.f b/SRC/ssytri2x.f index ea07f029..6e422a91 100644 --- a/SRC/ssytri2x.f +++ b/SRC/ssytri2x.f @@ -153,7 +153,7 @@ IF( UPPER ) THEN * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL STRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -266,7 +266,7 @@ END IF END DO * -* U11T*invD1*U11->U11 +* U11**T*invD1*U11->U11 * CALL STRMM('L','U','T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -277,12 +277,12 @@ END DO END DO * -* U01'invD*U01->A(CUT+I,CUT+J) +* U01**T*invD*U01->A(CUT+I,CUT+J) * CALL SGEMM('T','N',NNB,NNB,CUT,ONE,A(1,CUT+1),LDA, $ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* U11 = U11T*invD1*U11 + U01'invD*U01 +* U11 = U11**T*invD1*U11 + U01**T*invD*U01 * DO I=1,NNB DO J=I,NNB @@ -290,7 +290,7 @@ END DO END DO * -* U01 = U00T*invD0*U01 +* U01 = U00**T*invD0*U01 * CALL STRMM('L',UPLO,'T','U',CUT, NNB, $ ONE,A,LDA,WORK,N+NB+1) @@ -308,7 +308,7 @@ * END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=1 DO WHILE ( I .LE. N ) @@ -330,7 +330,7 @@ * * LOWER... * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL STRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -437,7 +437,7 @@ END IF END DO * -* L11T*invD1*L11->L11 +* L11**T*invD1*L11->L11 * CALL STRMM('L',UPLO,'T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -451,13 +451,13 @@ * IF ( (CUT+NNB) .LT. N ) THEN * -* L21T*invD2*L21->A(CUT+I,CUT+J) +* L21**T*invD2*L21->A(CUT+I,CUT+J) * CALL SGEMM('T','N',NNB,NNB,N-NNB-CUT,ONE,A(CUT+NNB+1,CUT+1) $ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* L11 = L11T*invD1*L11 + U01'invD*U01 +* L11 = L11**T*invD1*L11 + U01**T*invD*U01 * DO I=1,NNB DO J=1,I @@ -465,7 +465,7 @@ END DO END DO * -* L01 = L22T*invD2*L21 +* L01 = L22**T*invD2*L21 * CALL STRMM('L',UPLO,'T','U', N-NNB-CUT, NNB, $ ONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1) @@ -480,7 +480,7 @@ ELSE * -* L11 = L11T*invD1*L11 +* L11 = L11**T*invD1*L11 * DO I=1,NNB DO J=1,I @@ -494,7 +494,7 @@ CUT=CUT+NNB END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=N DO WHILE ( I .GE. 1 ) diff --git a/SRC/ssytrs2.f b/SRC/ssytrs2.f index 2091ac94..5d903212 100644 --- a/SRC/ssytrs2.f +++ b/SRC/ssytrs2.f @@ -118,7 +118,7 @@ * * Solve A*X = B, where A = U*D*U**T. * -* P' * B +* P**T * B K=N DO WHILE ( K .GE. 1 ) IF( IPIV( K ).GT.0 ) THEN @@ -195,7 +195,7 @@ * * Solve A*X = B, where A = L*D*L**T. * -* P' * B +* P**T * B K=1 DO WHILE ( K .LE. N ) IF( IPIV( K ).GT.0 ) THEN diff --git a/SRC/stbcon.f b/SRC/stbcon.f index 4e85f814..530d2271 100644 --- a/SRC/stbcon.f +++ b/SRC/stbcon.f @@ -170,7 +170,7 @@ $ AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**T). * CALL SLATBS( UPLO, 'Transpose', DIAG, NORMIN, N, KD, AB, $ LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO ) diff --git a/SRC/stbrfs.f b/SRC/stbrfs.f index 98a4fc9f..594444a7 100644 --- a/SRC/stbrfs.f +++ b/SRC/stbrfs.f @@ -195,7 +195,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL SCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL STBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), @@ -258,7 +258,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN IF( NOUNIT ) THEN diff --git a/SRC/stbtrs.f b/SRC/stbtrs.f index 688aebea..78fe9978 100644 --- a/SRC/stbtrs.f +++ b/SRC/stbtrs.f @@ -150,7 +150,7 @@ END IF INFO = 0 * -* Solve A * X = B or A' * X = B. +* Solve A * X = B or A**T * X = B. * DO 30 J = 1, NRHS CALL STBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, B( 1, J ), 1 ) diff --git a/SRC/stfsm.f b/SRC/stfsm.f index 3eee1f26..70b66ca4 100644 --- a/SRC/stfsm.f +++ b/SRC/stfsm.f @@ -31,7 +31,7 @@ * where alpha is a scalar, X and B are m by n matrices, A is a unit, or * non-unit, upper or lower triangular matrix and op( A ) is one of * -* op( A ) = A or op( A ) = A'. +* op( A ) = A or op( A ) = A**T. * * A is in Rectangular Full Packed (RFP) Format. * diff --git a/SRC/stgexc.f b/SRC/stgexc.f index 8c9c834d..71051bcb 100644 --- a/SRC/stgexc.f +++ b/SRC/stgexc.f @@ -21,7 +21,7 @@ * STGEXC reorders the generalized real Schur decomposition of a real * matrix pair (A,B) using an orthogonal equivalence transformation * -* (A, B) = Q * (A, B) * Z', +* (A, B) = Q * (A, B) * Z**T, * * so that the diagonal block of (A, B) with row index IFST is moved * to row ILST. diff --git a/SRC/stgsen.f b/SRC/stgsen.f index 356e986f..134cc83d 100644 --- a/SRC/stgsen.f +++ b/SRC/stgsen.f @@ -211,7 +211,7 @@ * ( 0 A22),( 0 B22) n2 * n1 n2 n1 n2 * -* where N = n1+n2 and U' means the transpose of U. The first n1 columns +* where N = n1+n2 and U**T means the transpose of U. The first n1 columns * of U and W span the specified pair of left and right eigenspaces * (deflating subspaces) of (A, B). * diff --git a/SRC/stgsja.f b/SRC/stgsja.f index af472d86..7818223f 100644 --- a/SRC/stgsja.f +++ b/SRC/stgsja.f @@ -249,7 +249,7 @@ * * U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1, * -* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose +* where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose * of Z. C1 and S1 are diagonal matrices satisfying * * C1**2 + S1**2 = I, diff --git a/SRC/stgsna.f b/SRC/stgsna.f index d3db5416..32759c67 100644 --- a/SRC/stgsna.f +++ b/SRC/stgsna.f @@ -25,7 +25,7 @@ * eigenvalues and/or eigenvectors of a matrix pair (A, B) in * generalized real Schur canonical form (or of any matrix pair * (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where -* Z' denotes the transpose of Z. +* Z**T denotes the transpose of Z. * * (A, B) must be in generalized real Schur form (as returned by SGGES), * i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal @@ -233,8 +233,8 @@ * and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an * upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) * -* Z2 = [ kron(S11', In-2) -kron(I2, S22) ] -* [ kron(T11', In-2) -kron(I2, T22) ] +* Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ] +* [ kron(T11**T, In-2) -kron(I2, T22) ] * * Note that if the default method for computing DIF is wanted (see * SLATDF), then the parameter DIFDRI (see below) should be changed diff --git a/SRC/stgsy2.f b/SRC/stgsy2.f index 20a4ae5e..74e3ba49 100644 --- a/SRC/stgsy2.f +++ b/SRC/stgsy2.f @@ -38,19 +38,19 @@ * In matrix notation solving equation (1) corresponds to solve * Z*x = scale*b, where Z is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ], +* Z = [ kron(In, A) -kron(B**T, Im) ] (2) +* [ kron(In, D) -kron(E**T, Im) ], * -* Ik is the identity matrix of size k and X' is the transpose of X. +* Ik is the identity matrix of size k and X**T is the transpose of X. * kron(X, Y) is the Kronecker product between the matrices X and Y. * In the process of solving (1), we solve a number of such systems * where Dim(In), Dim(In) = 1 or 2. * -* If TRANS = 'T', solve the transposed system Z'*y = scale*b for y, +* If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y, * which is equivalent to solve for R and L in * -* A' * R + D' * L = scale * C (3) -* R * B' + L * E' = scale * -F +* A**T * R + D**T * L = scale * C (3) +* R * B**T + L * E**T = scale * -F * * This case is used to compute an estimate of Dif[(A, D), (B, E)] = * sigma_min(Z) using reverse communicaton with SLACON. @@ -649,7 +649,7 @@ ZDIM = MB*NB*2 IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN * -* Build a 2-by-2 system Z' * x = RHS +* Build a 2-by-2 system Z**T * x = RHS * Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = -B( JS, JS ) @@ -661,7 +661,7 @@ RHS( 1 ) = C( IS, JS ) RHS( 2 ) = F( IS, JS ) * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL SGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) @@ -703,7 +703,7 @@ * ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN * -* Build a 4-by-4 system Z' * x = RHS +* Build a 4-by-4 system Z**T * x = RHS * Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = ZERO @@ -732,7 +732,7 @@ RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( IS, JSP1 ) * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL SGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) @@ -775,7 +775,7 @@ * ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN * -* Build a 4-by-4 system Z' * x = RHS +* Build a 4-by-4 system Z**T * x = RHS * Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = A( IS, ISP1 ) @@ -804,7 +804,7 @@ RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( ISP1, JS ) * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL SGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) @@ -846,7 +846,7 @@ * ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN * -* Build an 8-by-8 system Z' * x = RHS +* Build an 8-by-8 system Z**T * x = RHS * CALL SLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ ) * @@ -898,7 +898,7 @@ 160 CONTINUE * * -* Solve Z' * x = RHS +* Solve Z**T * x = RHS * CALL SGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) diff --git a/SRC/stgsyl.f b/SRC/stgsyl.f index f2c2a940..d1ec0b11 100644 --- a/SRC/stgsyl.f +++ b/SRC/stgsyl.f @@ -40,10 +40,10 @@ * In matrix notation (1) is equivalent to solve Zx = scale b, where * Z is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ]. +* Z = [ kron(In, A) -kron(B**T, Im) ] (2) +* [ kron(In, D) -kron(E**T, Im) ]. * -* Here Ik is the identity matrix of size k and X' is the transpose of +* Here Ik is the identity matrix of size k and X**T is the transpose of * X. kron(X, Y) is the Kronecker product between the matrices X and Y. * * If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b, diff --git a/SRC/stpcon.f b/SRC/stpcon.f index 31822b0b..6862d1f7 100644 --- a/SRC/stpcon.f +++ b/SRC/stpcon.f @@ -159,7 +159,7 @@ $ WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**T). * CALL SLATPS( UPLO, 'Transpose', DIAG, NORMIN, N, AP, $ WORK, SCALE, WORK( 2*N+1 ), INFO ) diff --git a/SRC/stprfs.f b/SRC/stprfs.f index c8bd6cda..2c913dc3 100644 --- a/SRC/stprfs.f +++ b/SRC/stprfs.f @@ -184,7 +184,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL SCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL STPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) @@ -250,7 +250,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN KC = 1 diff --git a/SRC/stptrs.f b/SRC/stptrs.f index 65e60d24..49f21a18 100644 --- a/SRC/stptrs.f +++ b/SRC/stptrs.f @@ -141,7 +141,7 @@ END IF INFO = 0 * -* Solve A * x = b or A' * x = b. +* Solve A * x = b or A**T * x = b. * DO 30 J = 1, NRHS CALL STPSV( UPLO, TRANS, DIAG, N, AP, B( 1, J ), 1 ) diff --git a/SRC/strcon.f b/SRC/strcon.f index 295481e9..9e93ee77 100644 --- a/SRC/strcon.f +++ b/SRC/strcon.f @@ -165,7 +165,7 @@ $ LDA, WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**T). * CALL SLATRS( UPLO, 'Transpose', DIAG, NORMIN, N, A, LDA, $ WORK, SCALE, WORK( 2*N+1 ), INFO ) diff --git a/SRC/strevc.f b/SRC/strevc.f index 82fe2fb4..12b1ae79 100644 --- a/SRC/strevc.f +++ b/SRC/strevc.f @@ -727,8 +727,8 @@ $ WORK( KI+1+N ), 1 ) * * Solve -* [T(J,J)-WR T(J,J+1) ]'* X = SCALE*( WORK1 ) -* [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 ) +* [T(J,J)-WR T(J,J+1) ]**T* X = SCALE*( WORK1 ) +* [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 ) * CALL SLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ), $ LDT, ONE, ONE, WORK( J+N ), N, WR, @@ -892,8 +892,8 @@ $ WORK( KI+2+N2 ), 1 ) * * Solve 2-by-2 complex linear equation -* ([T(j,j) T(j,j+1) ]'-(wr-i*wi)*I)*X = SCALE*B -* ([T(j+1,j) T(j+1,j+1)] ) +* ([T(j,j) T(j,j+1) ]**T-(wr-i*wi)*I)*X = SCALE*B +* ([T(j+1,j) T(j+1,j+1)] ) * CALL SLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ), $ LDT, ONE, ONE, WORK( J+N ), N, WR, diff --git a/SRC/strrfs.f b/SRC/strrfs.f index 42766a2b..d5fb484e 100644 --- a/SRC/strrfs.f +++ b/SRC/strrfs.f @@ -190,7 +190,7 @@ DO 250 J = 1, NRHS * * Compute residual R = B - op(A) * X, -* where op(A) = A or A', depending on TRANS. +* where op(A) = A or A**T, depending on TRANS. * CALL SCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL STRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 ) @@ -250,7 +250,7 @@ END IF ELSE * -* Compute abs(A')*abs(X) + abs(B). +* Compute abs(A**T)*abs(X) + abs(B). * IF( UPPER ) THEN IF( NOUNIT ) THEN diff --git a/SRC/strsen.f b/SRC/strsen.f index 724de085..a367fb2c 100644 --- a/SRC/strsen.f +++ b/SRC/strsen.f @@ -162,7 +162,7 @@ * ( 0 T22 ) n2 * n1 n2 * -* where N = n1+n2 and Z' means the transpose of Z. The first n1 columns +* where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns * of Z span the specified invariant subspace of T. * * If T has been obtained from the real Schur factorization of a matrix diff --git a/SRC/strtrs.f b/SRC/strtrs.f index 8dc42508..aaa70b92 100644 --- a/SRC/strtrs.f +++ b/SRC/strtrs.f @@ -136,7 +136,7 @@ END IF INFO = 0 * -* Solve A * x = b or A' * x = b. +* Solve A * x = b or A**T * x = b. * CALL STRSM( 'Left', UPLO, TRANS, DIAG, N, NRHS, ONE, A, LDA, B, $ LDB ) diff --git a/SRC/zgels.f b/SRC/zgels.f index e9bb64e0..1f1d25df 100644 --- a/SRC/zgels.f +++ b/SRC/zgels.f @@ -299,9 +299,9 @@ * ELSE * -* Overdetermined system of equations A' * X = B +* Overdetermined system of equations A**H * X = B * -* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) +* B(1:N,1:NRHS) := inv(R**H) * B(1:N,1:NRHS) * CALL ZTRTRS( 'Upper', 'Conjugate transpose','Non-unit', $ N, NRHS, A, LDA, B, LDB, INFO ) @@ -372,7 +372,7 @@ * ELSE * -* overdetermined system min || A' * X - B || +* overdetermined system min || A**H * X - B || * * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) * @@ -382,7 +382,7 @@ * * workspace at least NRHS, optimally NRHS*NB * -* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) +* B(1:M,1:NRHS) := inv(L**H) * B(1:M,1:NRHS) * CALL ZTRTRS( 'Lower', 'Conjugate transpose', 'Non-unit', $ M, NRHS, A, LDA, B, LDB, INFO ) diff --git a/SRC/zherfs.f b/SRC/zherfs.f index c09e044c..7564decf 100644 --- a/SRC/zherfs.f +++ b/SRC/zherfs.f @@ -308,7 +308,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/zhetd2.f b/SRC/zhetd2.f index 464702e6..4baf6981 100644 --- a/SRC/zhetd2.f +++ b/SRC/zhetd2.f @@ -237,7 +237,7 @@ CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**H +* A := A - v * w**H - w * v**H * CALL ZHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, $ A( I+1, I+1 ), LDA ) diff --git a/SRC/zhetrd.f b/SRC/zhetrd.f index f406b01e..e68d4e0b 100644 --- a/SRC/zhetrd.f +++ b/SRC/zhetrd.f @@ -237,7 +237,7 @@ $ LDWORK ) * * Update the unreduced submatrix A(1:i-1,1:i-1), using an -* update of the form: A := A - V*W' - W*V**H +* update of the form: A := A - V*W**H - W*V**H * CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE, $ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA ) @@ -268,7 +268,7 @@ $ TAU( I ), WORK, LDWORK ) * * Update the unreduced submatrix A(i+nb:n,i+nb:n), using -* an update of the form: A := A - V*W' - W*V**H +* an update of the form: A := A - V*W**H - W*V**H * CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE, $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, diff --git a/SRC/zhetri2x.f b/SRC/zhetri2x.f index 67ee4b96..6149a735 100644 --- a/SRC/zhetri2x.f +++ b/SRC/zhetri2x.f @@ -156,7 +156,7 @@ IF( UPPER ) THEN * -* invA = P * inv(U**H)*inv(D)*inv(U)*P'. +* invA = P * inv(U**H)*inv(D)*inv(U)*P**H. * CALL ZTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -269,7 +269,7 @@ END IF END DO * -* U11T*invD1*U11->U11 +* U11**H*invD1*U11->U11 * CALL ZTRMM('L','U','C','U',NNB, NNB, $ CONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -280,12 +280,12 @@ END DO END DO * -* U01'invD*U01->A(CUT+I,CUT+J) +* U01**H*invD*U01->A(CUT+I,CUT+J) * CALL ZGEMM('C','N',NNB,NNB,CUT,CONE,A(1,CUT+1),LDA, $ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* U11 = U11T*invD1*U11 + U01'invD*U01 +* U11 = U11**H*invD1*U11 + U01**H*invD*U01 * DO I=1,NNB DO J=I,NNB @@ -293,7 +293,7 @@ END DO END DO * -* U01 = U00T*invD0*U01 +* U01 = U00**H*invD0*U01 * CALL ZTRMM('L',UPLO,'C','U',CUT, NNB, $ CONE,A,LDA,WORK,N+NB+1) @@ -311,7 +311,7 @@ * END DO * -* Apply PERMUTATIONS P and P': P * inv(U**H)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**H: P * inv(U**H)*inv(D)*inv(U) *P**H * I=1 DO WHILE ( I .LE. N ) @@ -333,7 +333,7 @@ * * LOWER... * -* invA = P * inv(U**H)*inv(D)*inv(U)*P'. +* invA = P * inv(U**H)*inv(D)*inv(U)*P**H. * CALL ZTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -440,7 +440,7 @@ END IF END DO * -* L11T*invD1*L11->L11 +* L11**H*invD1*L11->L11 * CALL ZTRMM('L',UPLO,'C','U',NNB, NNB, $ CONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -453,13 +453,13 @@ * IF ( (CUT+NNB) .LT. N ) THEN * -* L21T*invD2*L21->A(CUT+I,CUT+J) +* L21**H*invD2*L21->A(CUT+I,CUT+J) * CALL ZGEMM('C','N',NNB,NNB,N-NNB-CUT,CONE,A(CUT+NNB+1,CUT+1) $ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* L11 = L11T*invD1*L11 + U01'invD*U01 +* L11 = L11**H*invD1*L11 + U01**H*invD*U01 * DO I=1,NNB DO J=1,I @@ -467,7 +467,7 @@ END DO END DO * -* L01 = L22T*invD2*L21 +* L01 = L22**H*invD2*L21 * CALL ZTRMM('L',UPLO,'C','U', N-NNB-CUT, NNB, $ CONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1) @@ -480,7 +480,7 @@ END DO ELSE * -* L11 = L11T*invD1*L11 +* L11 = L11**H*invD1*L11 * DO I=1,NNB DO J=1,I @@ -494,7 +494,7 @@ CUT=CUT+NNB END DO * -* Apply PERMUTATIONS P and P': P * inv(U**H)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**H: P * inv(U**H)*inv(D)*inv(U) *P**H * I=N DO WHILE ( I .GE. 1 ) diff --git a/SRC/zhfrk.f b/SRC/zhfrk.f index f85b75c5..d945ee3f 100644 --- a/SRC/zhfrk.f +++ b/SRC/zhfrk.f @@ -26,11 +26,11 @@ * * ZHFRK performs one of the Hermitian rank--k operations * -* C := alpha*A*conjg( A' ) + beta*C, +* C := alpha*A*A**H + beta*C, * * or * -* C := alpha*conjg( A' )*A + beta*C, +* C := alpha*A**H*A + beta*C, * * where alpha and beta are real scalars, C is an n--by--n Hermitian * matrix and A is an n--by--k matrix in the first case and a k--by--n @@ -60,9 +60,9 @@ * On entry, TRANS specifies the operation to be performed as * follows: * -* TRANS = 'N' or 'n' C := alpha*A*conjg( A' ) + beta*C. +* TRANS = 'N' or 'n' C := alpha*A*A**H + beta*C. * -* TRANS = 'C' or 'c' C := alpha*conjg( A' )*A + beta*C. +* TRANS = 'C' or 'c' C := alpha*A**H*A + beta*C. * * Unchanged on exit. * diff --git a/SRC/zhprfs.f b/SRC/zhprfs.f index d8c34a93..672d0234 100644 --- a/SRC/zhprfs.f +++ b/SRC/zhprfs.f @@ -306,7 +306,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/zhptrd.f b/SRC/zhptrd.f index 403bc4c4..140581a9 100644 --- a/SRC/zhptrd.f +++ b/SRC/zhptrd.f @@ -217,7 +217,7 @@ CALL ZAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: -* A := A - v * w' - w * v**H +* A := A - v * w**H - w * v**H * CALL ZHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, $ AP( I1I1 ) ) diff --git a/SRC/zla_gbrcond_c.f b/SRC/zla_gbrcond_c.f index 70765f39..67184c3d 100644 --- a/SRC/zla_gbrcond_c.f +++ b/SRC/zla_gbrcond_c.f @@ -219,7 +219,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/zla_gbrcond_x.f b/SRC/zla_gbrcond_x.f index 44cebbc2..08ff3266 100644 --- a/SRC/zla_gbrcond_x.f +++ b/SRC/zla_gbrcond_x.f @@ -202,7 +202,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/zla_gercond_c.f b/SRC/zla_gercond_c.f index 320f1be9..b6046a0c 100644 --- a/SRC/zla_gercond_c.f +++ b/SRC/zla_gercond_c.f @@ -195,7 +195,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/zla_gercond_x.f b/SRC/zla_gercond_x.f index 502e8be9..7b759a0e 100644 --- a/SRC/zla_gercond_x.f +++ b/SRC/zla_gercond_x.f @@ -178,7 +178,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/zla_hercond_c.f b/SRC/zla_hercond_c.f index 632dece1..d0531adb 100644 --- a/SRC/zla_hercond_c.f +++ b/SRC/zla_hercond_c.f @@ -204,7 +204,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/zla_hercond_x.f b/SRC/zla_hercond_x.f index 0b147f79..16f94d34 100644 --- a/SRC/zla_hercond_x.f +++ b/SRC/zla_hercond_x.f @@ -180,7 +180,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/zla_porcond_c.f b/SRC/zla_porcond_c.f index c3dfee1a..e2eee9d1 100644 --- a/SRC/zla_porcond_c.f +++ b/SRC/zla_porcond_c.f @@ -47,7 +47,7 @@ * * AF (input) COMPLEX*16 array, dimension (LDAF,N) * The triangular factor U or L from the Cholesky factorization -* A = U**T*U or A = L*L**T, as computed by ZPOTRF. +* A = U**H*U or A = L*L**H, as computed by ZPOTRF. * * LDAF (input) INTEGER * The leading dimension of the array AF. LDAF >= max(1,N). @@ -200,7 +200,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**H). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/zla_porcond_x.f b/SRC/zla_porcond_x.f index 9e0c0403..7ef63389 100644 --- a/SRC/zla_porcond_x.f +++ b/SRC/zla_porcond_x.f @@ -46,7 +46,7 @@ * * AF (input) COMPLEX*16 array, dimension (LDAF,N) * The triangular factor U or L from the Cholesky factorization -* A = U**T*U or A = L*L**T, as computed by ZPOTRF. +* A = U**H*U or A = L*L**H, as computed by ZPOTRF. * * LDAF (input) INTEGER * The leading dimension of the array AF. LDAF >= max(1,N). @@ -175,7 +175,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**H). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/zla_syrcond_c.f b/SRC/zla_syrcond_c.f index 9fc56485..42ef9db3 100644 --- a/SRC/zla_syrcond_c.f +++ b/SRC/zla_syrcond_c.f @@ -205,7 +205,7 @@ END IF ELSE * -* Multiply by inv(C'). +* Multiply by inv(C**T). * IF ( CAPPLY ) THEN DO I = 1, N diff --git a/SRC/zla_syrcond_x.f b/SRC/zla_syrcond_x.f index 51c24cae..aaac999e 100644 --- a/SRC/zla_syrcond_x.f +++ b/SRC/zla_syrcond_x.f @@ -181,7 +181,7 @@ END DO ELSE * -* Multiply by inv(X'). +* Multiply by inv(X**T). * DO I = 1, N WORK( I ) = WORK( I ) / X( I ) diff --git a/SRC/zlacn2.f b/SRC/zlacn2.f index f099e853..b3564089 100644 --- a/SRC/zlacn2.f +++ b/SRC/zlacn2.f @@ -33,8 +33,8 @@ * X (input/output) COMPLEX*16 array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, -* where A' is the conjugate transpose of A, and ZLACN2 must be +* A**H * X, if KASE=2, +* where A**H is the conjugate transpose of A, and ZLACN2 must be * re-called with all the other parameters unchanged. * * EST (input/output) DOUBLE PRECISION @@ -45,7 +45,7 @@ * KASE (input/output) INTEGER * On the initial call to ZLACN2, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**H * X. * On the final return from ZLACN2, KASE will again be 0. * * ISAVE (input/output) INTEGER array, dimension (3) diff --git a/SRC/zlacon.f b/SRC/zlacon.f index 46bfeb57..d49f1574 100644 --- a/SRC/zlacon.f +++ b/SRC/zlacon.f @@ -32,8 +32,8 @@ * X (input/output) COMPLEX*16 array, dimension (N) * On an intermediate return, X should be overwritten by * A * X, if KASE=1, -* A' * X, if KASE=2, -* where A' is the conjugate transpose of A, and ZLACON must be +* A**H * X, if KASE=2, +* where A**H is the conjugate transpose of A, and ZLACON must be * re-called with all the other parameters unchanged. * * EST (input/output) DOUBLE PRECISION @@ -44,7 +44,7 @@ * KASE (input/output) INTEGER * On the initial call to ZLACON, KASE should be 0. * On an intermediate return, KASE will be 1 or 2, indicating -* whether X should be overwritten by A * X or A' * X. +* whether X should be overwritten by A * X or A**H * X. * On the final return from ZLACON, KASE will again be 0. * * Further Details diff --git a/SRC/zlaesy.f b/SRC/zlaesy.f index 439dfa0e..12d0904f 100644 --- a/SRC/zlaesy.f +++ b/SRC/zlaesy.f @@ -127,7 +127,7 @@ * * Choose CS1 = 1 and SN1 to satisfy the first equation, then * scale the components of this eigenvector so that the matrix -* of eigenvectors X satisfies X * X' = I . (No scaling is +* of eigenvectors X satisfies X * X**T = I . (No scaling is * done if the norm of the eigenvalue matrix is less than THRESH.) * SN1 = ( RT1-A ) / B diff --git a/SRC/zlahr2.f b/SRC/zlahr2.f index db310900..b670de14 100644 --- a/SRC/zlahr2.f +++ b/SRC/zlahr2.f @@ -151,7 +151,7 @@ $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 ) CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) * -* Apply I - V * T' * V**H to this column (call it b) from the +* Apply I - V * T**H * V**H to this column (call it b) from the * left, using the last column of T as workspace * * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) @@ -159,7 +159,7 @@ * * where V1 is unit lower triangular * -* w := V1' * b1 +* w := V1**H * b1 * CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT', diff --git a/SRC/zlaic1.f b/SRC/zlaic1.f index 5b73020d..4e5230f6 100644 --- a/SRC/zlaic1.f +++ b/SRC/zlaic1.f @@ -28,15 +28,15 @@ * [ s*x ] * xhat = [ c ] * is an approximate singular vector of -* [ L 0 ] -* Lhat = [ w' gamma ] +* [ L 0 ] +* Lhat = [ w**H gamma ] * in the sense that * twonorm(Lhat*xhat) = sestpr. * * Depending on JOB, an estimate for the largest or smallest singular * value is computed. * -* Note that [s c]' and sestpr**2 is an eigenpair of the system +* Note that [s c]**H and sestpr**2 is an eigenpair of the system * * diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ] * [ conjg(gamma) ] diff --git a/SRC/zlalsa.f b/SRC/zlalsa.f index 41ff70d6..eee6476b 100644 --- a/SRC/zlalsa.f +++ b/SRC/zlalsa.f @@ -79,7 +79,7 @@ * POLES, GIVNUM, and Z. * * VT (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ). -* On entry, VT' contains the right singular vector matrices of +* On entry, VT**H contains the right singular vector matrices of * all subproblems at the bottom level. * * K (input) INTEGER array, dimension ( N ). diff --git a/SRC/zlalsd.f b/SRC/zlalsd.f index 24cf5515..2eee0686 100644 --- a/SRC/zlalsd.f +++ b/SRC/zlalsd.f @@ -290,7 +290,7 @@ * * Since B is complex, the following call to DGEMM is performed * in two steps (real and imaginary parts). That is for V * B -* (in the real version of the code V' is stored in WORK). +* (in the real version of the code V**H is stored in WORK). * * CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, * $ WORK( NWORK ), N ) diff --git a/SRC/zlanhf.f b/SRC/zlanhf.f index ae5ddff4..60da0db2 100644 --- a/SRC/zlanhf.f +++ b/SRC/zlanhf.f @@ -1026,7 +1026,7 @@ ELSE * A is xpose & A is k by n IF( ILU.EQ.0 ) THEN -* A' is upper +* A**H is upper DO J = 1, K - 2 CALL ZLASSQ( J, A( 0+( K+J )*LDA ), 1, SCALE, S ) * U at A(0,k) @@ -1080,7 +1080,7 @@ L = L + LDA + 1 END DO ELSE -* A' is lower +* A**H is lower DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+J*LDA ), 1, SCALE, S ) * U at A(0,0) @@ -1215,7 +1215,7 @@ ELSE * A is xpose IF( ILU.EQ.0 ) THEN -* A' is upper +* A**H is upper DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+( K+1+J )*LDA ), 1, SCALE, S ) * U at A(0,k+1) @@ -1281,7 +1281,7 @@ END IF END IF ELSE -* A' is lower +* A**H is lower DO J = 1, K - 1 CALL ZLASSQ( J, A( 0+( J+1 )*LDA ), 1, SCALE, S ) * U at A(0,1) diff --git a/SRC/zlaqr5.f b/SRC/zlaqr5.f index 0e4fb1e0..15c8b113 100644 --- a/SRC/zlaqr5.f +++ b/SRC/zlaqr5.f @@ -635,14 +635,14 @@ CALL ZLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ), $ LDH, WH( KZS+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**H ==== * CALL ZLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH ) CALL ZTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE, $ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ), $ LDWH ) * -* ==== Multiply top of H by U11' ==== +* ==== Multiply top of H by U11**H ==== * CALL ZGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU, $ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH ) @@ -652,7 +652,7 @@ CALL ZLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH, $ WH( I2+1, 1 ), LDWH ) * -* ==== Multiply by U21' ==== +* ==== Multiply by U21**H ==== * CALL ZTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE, $ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH ) diff --git a/SRC/zlarfb.f b/SRC/zlarfb.f index a2b93be8..075e3b67 100644 --- a/SRC/zlarfb.f +++ b/SRC/zlarfb.f @@ -289,7 +289,7 @@ LASTV = MAX( K, ILAZLR( M, K, V, LDV ) ) LASTC = ILAZLC( LASTV, N, C, LDC ) * -* W := C' * V = (C1**H * V1 + C2**H * V2) (stored in WORK) +* W := C**H * V = (C1**H * V1 + C2**H * V2) (stored in WORK) * * W := C2**H * diff --git a/SRC/zlarzt.f b/SRC/zlarzt.f index 2ae745ee..e25a3376 100644 --- a/SRC/zlarzt.f +++ b/SRC/zlarzt.f @@ -27,12 +27,12 @@ * If STOREV = 'C', the vector which defines the elementary reflector * H(i) is stored in the i-th column of the array V, and * -* H = I - V * T * V' +* H = I - V * T * V**H * * If STOREV = 'R', the vector which defines the elementary reflector * H(i) is stored in the i-th row of the array V, and * -* H = I - V' * T * V +* H = I - V**H * T * V * * Currently, only STOREV = 'R' and DIRECT = 'B' are supported. * diff --git a/SRC/zlasr.f b/SRC/zlasr.f index 38ff3970..7e665cf1 100644 --- a/SRC/zlasr.f +++ b/SRC/zlasr.f @@ -274,7 +274,7 @@ END IF ELSE IF( LSAME( SIDE, 'R' ) ) THEN * -* Form A * P' +* Form A * P**T * IF( LSAME( PIVOT, 'V' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN diff --git a/SRC/zlatbs.f b/SRC/zlatbs.f index d15295f1..7f467df0 100644 --- a/SRC/zlatbs.f +++ b/SRC/zlatbs.f @@ -24,7 +24,7 @@ * A * x = s*b, A**T * x = s*b, or A**H * x = s*b, * * with scaling to prevent overflow, where A is an upper or lower -* triangular band matrix. Here A' denotes the transpose of A, x and b +* triangular band matrix. Here A**T denotes the transpose of A, x and b * are n-element vectors, and s is a scaling factor, usually less than * or equal to 1, chosen so that the components of x will be less than * the overflow threshold. If the unscaled problem will not cause diff --git a/SRC/zpbrfs.f b/SRC/zpbrfs.f index d1b0cd23..254c8bc8 100644 --- a/SRC/zpbrfs.f +++ b/SRC/zpbrfs.f @@ -311,7 +311,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/zpbstf.f b/SRC/zpbstf.f index 1afad775..b63fba71 100644 --- a/SRC/zpbstf.f +++ b/SRC/zpbstf.f @@ -83,20 +83,21 @@ * * on entry: on exit: * -* * * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75' -* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76' -* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 +* * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H +* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H +* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 * * If UPLO = 'L', the array AB holds: * * on entry: on exit: * -* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 -* a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * -* a31 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * * +* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 +* a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 * +* a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * * * -* Array elements marked * are not used by the routine; s12' denotes +* Array elements marked * are not used by the routine; s12**H denotes * conjg(s12); the diagonal elements of S are real. + * * ===================================================================== * diff --git a/SRC/zporfs.f b/SRC/zporfs.f index ac864758..ad242d7e 100644 --- a/SRC/zporfs.f +++ b/SRC/zporfs.f @@ -302,7 +302,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/zpprfs.f b/SRC/zpprfs.f index 07dd64cf..99d3a2f7 100644 --- a/SRC/zpprfs.f +++ b/SRC/zpprfs.f @@ -300,7 +300,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**H). * CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/zppsvx.f b/SRC/zppsvx.f index c2cd22f4..27e17a34 100644 --- a/SRC/zppsvx.f +++ b/SRC/zppsvx.f @@ -46,7 +46,7 @@ * A = U**H * U , if UPLO = 'U', or * A = L * L**H, if UPLO = 'L', * where U is an upper triangular matrix, L is a lower triangular -* matrix, and ' indicates conjugate transpose. +* matrix, and **H indicates conjugate transpose. * * 3. If the leading i-by-i principal minor is not positive definite, * then the routine returns with INFO = i. Otherwise, the factored diff --git a/SRC/zpstf2.f b/SRC/zpstf2.f index da8c9cd7..568980b7 100644 --- a/SRC/zpstf2.f +++ b/SRC/zpstf2.f @@ -22,8 +22,8 @@ * pivoting of a complex Hermitian positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**H * U , if UPLO = 'U', +* P**T * A * P = L * L**H, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -162,7 +162,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**H* U * DO 150 J = 1, N * @@ -234,7 +234,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**H * DO 180 J = 1, N * diff --git a/SRC/zpstrf.f b/SRC/zpstrf.f index 9bc152ee..90970cd7 100644 --- a/SRC/zpstrf.f +++ b/SRC/zpstrf.f @@ -25,8 +25,8 @@ * pivoting of a complex Hermitian positive semidefinite matrix A. * * The factorization has the form -* P' * A * P = U' * U , if UPLO = 'U', -* P' * A * P = L * L', if UPLO = 'L', +* P**T * A * P = U**H * U , if UPLO = 'U', +* P**T * A * P = L * L**H, if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular, and * P is stored as vector PIV. * @@ -175,7 +175,7 @@ * IF( UPPER ) THEN * -* Compute the Cholesky factorization P' * A * P = U' * U +* Compute the Cholesky factorization P**T * A * P = U**H * U * DO 160 K = 1, N, NB * @@ -270,7 +270,7 @@ * ELSE * -* Compute the Cholesky factorization P' * A * P = L * L' +* Compute the Cholesky factorization P**T * A * P = L * L**H * DO 210 K = 1, N, NB * diff --git a/SRC/zptcon.f b/SRC/zptcon.f index 4c7992b0..dbee4199 100644 --- a/SRC/zptcon.f +++ b/SRC/zptcon.f @@ -118,7 +118,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**H. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H. * * Solve M(L) * x = e. * diff --git a/SRC/zptrfs.f b/SRC/zptrfs.f index 90577ca9..c3ae8f23 100644 --- a/SRC/zptrfs.f +++ b/SRC/zptrfs.f @@ -327,7 +327,7 @@ * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * -* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)**H. +* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H. * * Solve M(L) * x = e. * @@ -19,7 +19,7 @@ * * ZSPR performs the symmetric rank 1 operation * -* A := alpha*x*conjg( x' ) + A, +* A := alpha*x*x**H + A, * * where alpha is a complex scalar, x is an n element vector and A is an * n by n symmetric matrix, supplied in packed form. diff --git a/SRC/zsprfs.f b/SRC/zsprfs.f index e161062c..cb473d05 100644 --- a/SRC/zsprfs.f +++ b/SRC/zsprfs.f @@ -305,7 +305,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) DO 110 I = 1, N @@ -19,7 +19,7 @@ * * ZSYR performs the symmetric rank 1 operation * -* A := alpha*x*( x' ) + A, +* A := alpha*x*x**H + A, * * where alpha is a complex scalar, x is an n element vector and A is an * n by n symmetric matrix. diff --git a/SRC/zsyrfs.f b/SRC/zsyrfs.f index b4cb47ea..2609604a 100644 --- a/SRC/zsyrfs.f +++ b/SRC/zsyrfs.f @@ -308,7 +308,7 @@ IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * -* Multiply by diag(W)*inv(A'). +* Multiply by diag(W)*inv(A**T). * CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) DO 110 I = 1, N diff --git a/SRC/zsytri2x.f b/SRC/zsytri2x.f index c39e4175..e2ba02ff 100644 --- a/SRC/zsytri2x.f +++ b/SRC/zsytri2x.f @@ -154,7 +154,7 @@ IF( UPPER ) THEN * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL ZTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -267,7 +267,7 @@ END IF END DO * -* U11T*invD1*U11->U11 +* U11**T*invD1*U11->U11 * CALL ZTRMM('L','U','T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -278,12 +278,12 @@ END DO END DO * -* U01'invD*U01->A(CUT+I,CUT+J) +* U01**T*invD*U01->A(CUT+I,CUT+J) * CALL ZGEMM('T','N',NNB,NNB,CUT,ONE,A(1,CUT+1),LDA, $ WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* U11 = U11T*invD1*U11 + U01'invD*U01 +* U11 = U11**T*invD1*U11 + U01**T*invD*U01 * DO I=1,NNB DO J=I,NNB @@ -291,7 +291,7 @@ END DO END DO * -* U01 = U00T*invD0*U01 +* U01 = U00**T*invD0*U01 * CALL ZTRMM('L',UPLO,'T','U',CUT, NNB, $ ONE,A,LDA,WORK,N+NB+1) @@ -309,7 +309,7 @@ * END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=1 DO WHILE ( I .LE. N ) @@ -331,7 +331,7 @@ * * LOWER... * -* invA = P * inv(U**T)*inv(D)*inv(U)*P'. +* invA = P * inv(U**T)*inv(D)*inv(U)*P**T. * CALL ZTRTRI( UPLO, 'U', N, A, LDA, INFO ) * @@ -438,7 +438,7 @@ END IF END DO * -* L11T*invD1*L11->L11 +* L11**T*invD1*L11->L11 * CALL ZTRMM('L',UPLO,'T','U',NNB, NNB, $ ONE,A(CUT+1,CUT+1),LDA,WORK(U11+1,1),N+NB+1) @@ -452,13 +452,13 @@ IF ( (CUT+NNB) .LT. N ) THEN * -* L21T*invD2*L21->A(CUT+I,CUT+J) +* L21**T*invD2*L21->A(CUT+I,CUT+J) * CALL ZGEMM('T','N',NNB,NNB,N-NNB-CUT,ONE,A(CUT+NNB+1,CUT+1) $ ,LDA,WORK,N+NB+1, ZERO, WORK(U11+1,1), N+NB+1) * -* L11 = L11T*invD1*L11 + U01'invD*U01 +* L11 = L11**T*invD1*L11 + U01**T*invD*U01 * DO I=1,NNB DO J=1,I @@ -466,7 +466,7 @@ END DO END DO * -* U01 = L22T*invD2*L21 +* U01 = L22**T*invD2*L21 * CALL ZTRMM('L',UPLO,'T','U', N-NNB-CUT, NNB, $ ONE,A(CUT+NNB+1,CUT+NNB+1),LDA,WORK,N+NB+1) @@ -479,7 +479,7 @@ END DO ELSE * -* L11 = L11T*invD1*L11 +* L11 = L11**T*invD1*L11 * DO I=1,NNB DO J=1,I @@ -493,7 +493,7 @@ CUT=CUT+NNB END DO * -* Apply PERMUTATIONS P and P': P * inv(U**T)*inv(D)*inv(U) *P' +* Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T * I=N DO WHILE ( I .GE. 1 ) diff --git a/SRC/ztbcon.f b/SRC/ztbcon.f index 1ed55654..5bd36c35 100644 --- a/SRC/ztbcon.f +++ b/SRC/ztbcon.f @@ -148,7 +148,7 @@ RCOND = ZERO SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( N, 1 ) ) * -* Compute the 1-norm of the triangular matrix A or A'. +* Compute the 1-norm of the triangular matrix A or A**H. * ANORM = ZLANTB( NORM, UPLO, DIAG, N, KD, AB, LDAB, RWORK ) * @@ -177,7 +177,7 @@ $ AB, LDAB, WORK, SCALE, RWORK, INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**H). * CALL ZLATBS( UPLO, 'Conjugate transpose', DIAG, NORMIN, $ N, KD, AB, LDAB, WORK, SCALE, RWORK, INFO ) diff --git a/SRC/ztfsm.f b/SRC/ztfsm.f index e40fbf78..803d68ae 100644 --- a/SRC/ztfsm.f +++ b/SRC/ztfsm.f @@ -31,7 +31,7 @@ * where alpha is a scalar, X and B are m by n matrices, A is a unit, or * non-unit, upper or lower triangular matrix and op( A ) is one of * -* op( A ) = A or op( A ) = conjg( A' ). +* op( A ) = A or op( A ) = A**H. * * A is in Rectangular Full Packed (RFP) Format. * diff --git a/SRC/ztgexc.f b/SRC/ztgexc.f index d5668359..fa853c54 100644 --- a/SRC/ztgexc.f +++ b/SRC/ztgexc.f @@ -20,7 +20,7 @@ * * ZTGEXC reorders the generalized Schur decomposition of a complex * matrix pair (A,B), using an unitary equivalence transformation -* (A, B) := Q * (A, B) * Z', so that the diagonal block of (A, B) with +* (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with * row index IFST is moved to row ILST. * * (A, B) must be in generalized Schur canonical form, that is, A and diff --git a/SRC/ztgsen.f b/SRC/ztgsen.f index 70a0d48d..bb11ae65 100644 --- a/SRC/ztgsen.f +++ b/SRC/ztgsen.f @@ -231,11 +231,11 @@ * where sigma-min(Zu) is the smallest singular value of the * (2*n1*n2)-by-(2*n1*n2) matrix * -* Zu = [ kron(In2, A11) -kron(A22', In1) ] -* [ kron(In2, B11) -kron(B22', In1) ]. +* Zu = [ kron(In2, A11) -kron(A22**H, In1) ] +* [ kron(In2, B11) -kron(B22**H, In1) ]. * -* Here, Inx is the identity matrix of size nx and A22' is the -* transpose of A22. kron(X, Y) is the Kronecker product between +* Here, Inx is the identity matrix of size nx and A22**H is the +* conjugate transpose of A22. kron(X, Y) is the Kronecker product between * the matrices X and Y. * * When DIF(2) is small, small changes in (A, B) can cause large changes diff --git a/SRC/ztgsy2.f b/SRC/ztgsy2.f index f91363ad..52f4a0df 100644 --- a/SRC/ztgsy2.f +++ b/SRC/ztgsy2.f @@ -22,7 +22,7 @@ * * ZTGSY2 solves the generalized Sylvester equation * -* A * R - L * B = scale * C (1) +* A * R - L * B = scale * C (1) * D * R - L * E = scale * F * * using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices, @@ -36,17 +36,17 @@ * In matrix notation solving equation (1) corresponds to solve * Zx = scale * b, where Z is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ], +* Z = [ kron(In, A) -kron(B**H, Im) ] (2) +* [ kron(In, D) -kron(E**H, Im) ], * -* Ik is the identity matrix of size k and X' is the transpose of X. +* Ik is the identity matrix of size k and X**H is the conjuguate transpose of X. * kron(X, Y) is the Kronecker product between the matrices X and Y. * -* If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b +* If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b * is solved for, which is equivalent to solve for R and L in * -* A' * R + D' * L = scale * C (3) -* R * B' + L * E' = scale * -F +* A**H * R + D**H * L = scale * C (3) +* R * B**H + L * E**H = scale * -F * * This case is used to compute an estimate of Dif[(A, D), (B, E)] = * = sigma_min(Z) using reverse communicaton with ZLACON. @@ -307,7 +307,7 @@ DO 80 I = 1, M DO 70 J = N, 1, -1 * -* Build 2 by 2 system Z' +* Build 2 by 2 system Z**H * Z( 1, 1 ) = DCONJG( A( I, I ) ) Z( 2, 1 ) = -DCONJG( B( J, J ) ) @@ -320,7 +320,7 @@ RHS( 1 ) = C( I, J ) RHS( 2 ) = F( I, J ) * -* Solve Z' * x = RHS +* Solve Z**H * x = RHS * CALL ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) diff --git a/SRC/ztgsyl.f b/SRC/ztgsyl.f index 7548bbbd..aedaf182 100644 --- a/SRC/ztgsyl.f +++ b/SRC/ztgsyl.f @@ -39,10 +39,10 @@ * In matrix notation (1) is equivalent to solve Zx = scale*b, where Z * is defined as * -* Z = [ kron(In, A) -kron(B', Im) ] (2) -* [ kron(In, D) -kron(E', Im) ], +* Z = [ kron(In, A) -kron(B**H, Im) ] (2) +* [ kron(In, D) -kron(E**H, Im) ], * -* Here Ix is the identity matrix of size x and X' is the conjugate +* Here Ix is the identity matrix of size x and X**H is the conjugate * transpose of X. Kron(X, Y) is the Kronecker product between the * matrices X and Y. * diff --git a/SRC/ztpcon.f b/SRC/ztpcon.f index ddb7e1f0..a48990fc 100644 --- a/SRC/ztpcon.f +++ b/SRC/ztpcon.f @@ -166,7 +166,7 @@ $ WORK, SCALE, RWORK, INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**H). * CALL ZLATPS( UPLO, 'Conjugate transpose', DIAG, NORMIN, $ N, AP, WORK, SCALE, RWORK, INFO ) diff --git a/SRC/ztrcon.f b/SRC/ztrcon.f index 8b173cdf..6dc12f7a 100644 --- a/SRC/ztrcon.f +++ b/SRC/ztrcon.f @@ -172,7 +172,7 @@ $ LDA, WORK, SCALE, RWORK, INFO ) ELSE * -* Multiply by inv(A'). +* Multiply by inv(A**H). * CALL ZLATRS( UPLO, 'Conjugate transpose', DIAG, NORMIN, $ N, A, LDA, WORK, SCALE, RWORK, INFO ) diff --git a/SRC/ztrttf.f b/SRC/ztrttf.f index fdc626ca..f1103a06 100644 --- a/SRC/ztrttf.f +++ b/SRC/ztrttf.f @@ -72,15 +72,15 @@ * 55 50 51 52 53 54 55 * * -* Let TRANSR = `N'. RFP holds AP as follows: -* For UPLO = `U' the upper trapezoid A(0:5,0:2) consists of the last +* Let TRANSR = 'N'. RFP holds AP as follows: +* For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last * three columns of AP upper. The lower triangle A(4:6,0:2) consists of * conjugate-transpose of the first three columns of AP upper. -* For UPLO = `L' the lower trapezoid A(1:6,0:2) consists of the first +* For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first * three columns of AP lower. The upper triangle A(0:2,0:2) consists of * conjugate-transpose of the last three columns of AP lower. * To denote conjugate we place -- above the element. This covers the -* case N even and TRANSR = `N'. +* case N even and TRANSR = 'N'. * * RFP A RFP A * @@ -99,7 +99,7 @@ * -- -- -- * 02 12 22 50 51 52 * -* Now let TRANSR = `C'. RFP A in both UPLO cases is just the conjugate- +* Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- * transpose of RFP A above. One therefore gets: * * @@ -125,15 +125,15 @@ * 44 40 41 42 43 44 * * -* Let TRANSR = `N'. RFP holds AP as follows: -* For UPLO = `U' the upper trapezoid A(0:4,0:2) consists of the last +* Let TRANSR = 'N'. RFP holds AP as follows: +* For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last * three columns of AP upper. The lower triangle A(3:4,0:1) consists of * conjugate-transpose of the first two columns of AP upper. -* For UPLO = `L' the lower trapezoid A(0:4,0:2) consists of the first +* For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first * three columns of AP lower. The upper triangle A(0:1,1:2) consists of * conjugate-transpose of the last two columns of AP lower. * To denote conjugate we place -- above the element. This covers the -* case N odd and TRANSR = `N'. +* case N odd and TRANSR = 'N'. * * RFP A RFP A * @@ -148,7 +148,7 @@ * -- -- * 01 11 44 40 41 42 * -* Now let TRANSR = `C'. RFP A in both UPLO cases is just the conjugate- +* Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- * transpose of RFP A above. One therefore gets: * * diff --git a/SRC/zungbr.f b/SRC/zungbr.f index 6d372212..c918ee3f 100644 --- a/SRC/zungbr.f +++ b/SRC/zungbr.f @@ -202,7 +202,7 @@ END IF ELSE * -* Form P', determined by a call to ZGEBRD to reduce a k-by-n +* Form P**H, determined by a call to ZGEBRD to reduce a k-by-n * matrix * IF( K.LT.N ) THEN @@ -216,7 +216,7 @@ * If k >= n, assume m = n * * Shift the vectors which define the elementary reflectors one -* row downward, and set the first row and column of P' to +* row downward, and set the first row and column of P**H to * those of the unit matrix * A( 1, 1 ) = ONE @@ -231,7 +231,7 @@ 60 CONTINUE IF( N.GT.1 ) THEN * -* Form P'(2:n,2:n) +* Form P**H(2:n,2:n) * CALL ZUNGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK, $ LWORK, IINFO ) |