diff options
| -rw-r--r-- | SRC/chetf2_rook.f | 910 | ||||
| -rw-r--r-- | SRC/zhetf2_rook.f | 910 |
2 files changed, 1820 insertions, 0 deletions
diff --git a/SRC/chetf2_rook.f b/SRC/chetf2_rook.f new file mode 100644 index 00000000..2404929c --- /dev/null +++ b/SRC/chetf2_rook.f @@ -0,0 +1,910 @@ +*> \brief \b CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download CHETF2_ROOK + dependencies +*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chetf2_rook.f"> +*> [TGZ]</a> +*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chetf2_rook.f"> +*> [ZIP]</a> +*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chetf2_rook.f"> +*> [TXT]</a> +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE CHETF2_ROOK( UPLO, N, A, LDA, IPIV, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER UPLO +* INTEGER INFO, LDA, N +* .. +* .. Array Arguments .. +* INTEGER IPIV( * ) +* COMPLEX A( LDA, * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> CHETF2_ROOK computes the factorization of a complex Hermitian matrix A +*> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method: +*> +*> A = U*D*U**H or A = L*D*L**H +*> +*> where U (or L) is a product of permutation and unit upper (lower) +*> triangular matrices, U**H is the conjugate transpose of U, and D is +*> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. +*> +*> This is the unblocked version of the algorithm, calling Level 2 BLAS. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] UPLO +*> \verbatim +*> UPLO is CHARACTER*1 +*> Specifies whether the upper or lower triangular part of the +*> Hermitian matrix A is stored: +*> = 'U': Upper triangular +*> = 'L': Lower triangular +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX array, dimension (LDA,N) +*> On entry, the Hermitian matrix A. If UPLO = 'U', the leading +*> n-by-n upper triangular part of A contains the upper +*> triangular part of the matrix A, and the strictly lower +*> triangular part of A is not referenced. If UPLO = 'L', the +*> leading n-by-n lower triangular part of A contains the lower +*> triangular part of the matrix A, and the strictly upper +*> triangular part of A is not referenced. +*> +*> On exit, the block diagonal matrix D and the multipliers used +*> to obtain the factor U or L (see below for further details). +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,N). +*> \endverbatim +*> +*> \param[out] IPIV +*> \verbatim +*> IPIV is INTEGER array, dimension (N) +*> Details of the interchanges and the block structure of D. +*> +*> If UPLO = 'U': +*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were +*> interchanged and D(k,k) is a 1-by-1 diagonal block. +*> +*> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and +*> columns k and -IPIV(k) were interchanged and rows and +*> columns k-1 and -IPIV(k-1) were inerchaged, +*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. +*> +*> If UPLO = 'L': +*> If IPIV(k) > 0, then rows and columns k and IPIV(k) +*> were interchanged and D(k,k) is a 1-by-1 diagonal block. +*> +*> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and +*> columns k and -IPIV(k) were interchanged and rows and +*> columns k+1 and -IPIV(k+1) were inerchaged, +*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -k, the k-th argument had an illegal value +*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization +*> has been completed, but the block diagonal matrix D is +*> exactly singular, and division by zero will occur if it +*> is used to solve a system of equations. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2012 +* +*> \ingroup complexHEcomputational +* +*> \par Further Details: +* ===================== +*> +*> \verbatim +*> +*> If UPLO = 'U', then A = U*D*U**H, where +*> U = P(n)*U(n)* ... *P(k)U(k)* ..., +*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to +*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 +*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as +*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such +*> that if the diagonal block D(k) is of order s (s = 1 or 2), then +*> +*> ( I v 0 ) k-s +*> U(k) = ( 0 I 0 ) s +*> ( 0 0 I ) n-k +*> k-s s n-k +*> +*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). +*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), +*> and A(k,k), and v overwrites A(1:k-2,k-1:k). +*> +*> If UPLO = 'L', then A = L*D*L**H, where +*> L = P(1)*L(1)* ... *P(k)*L(k)* ..., +*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to +*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 +*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as +*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such +*> that if the diagonal block D(k) is of order s (s = 1 or 2), then +*> +*> ( I 0 0 ) k-1 +*> L(k) = ( 0 I 0 ) s +*> ( 0 v I ) n-k-s+1 +*> k-1 s n-k-s+1 +*> +*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). +*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), +*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). +*> \endverbatim +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2012, Igor Kozachenko, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, +*> School of Mathematics, +*> University of Manchester +*> +*> 01-01-96 - Based on modifications by +*> J. Lewis, Boeing Computer Services Company +*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA +*> \endverbatim +* +* ===================================================================== + SUBROUTINE CHETF2_ROOK( UPLO, N, A, LDA, IPIV, INFO ) +* +* -- LAPACK computational routine (version 3.4.2) -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* November 2012 +* +* .. Scalar Arguments .. + CHARACTER UPLO + INTEGER INFO, LDA, N +* .. +* .. Array Arguments .. + INTEGER IPIV( * ) + COMPLEX A( LDA, * ) +* .. +* +* ====================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) + REAL EIGHT, SEVTEN + PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 ) +* .. +* .. Local Scalars .. + LOGICAL DONE, UPPER + INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP, + $ P + REAL ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, STEMP, + $ ROWMAX, TT, SFMIN + COMPLEX D12, D21, T, WK, WKM1, WKP1, Z +* .. +* .. External Functions .. +* + LOGICAL LSAME + INTEGER ICAMAX + REAL SLAMCH, SLAPY2 + EXTERNAL LSAME, ICAMAX, SLAMCH, SLAPY2 +* .. +* .. External Subroutines .. + EXTERNAL XERBLA, CSSCAL, CHER, CSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, REAL, SQRT +* .. +* .. Statement Functions .. + REAL CABS1 +* .. +* .. Statement Function definitions .. + CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) ) +* .. +* .. Executable Statements .. +* +* Test the input parameters. +* + INFO = 0 + UPPER = LSAME( UPLO, 'U' ) + IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -4 + END IF + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'CHETF2_ROOK', -INFO ) + RETURN + END IF +* +* Initialize ALPHA for use in choosing pivot block size. +* + ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT +* +* Compute machine safe minimum +* + SFMIN = SLAMCH( 'S' ) +* + IF( UPPER ) THEN +* +* Factorize A as U*D*U**H using the upper triangle of A +* +* K is the main loop index, decreasing from N to 1 in steps of +* 1 or 2 +* + K = N + 10 CONTINUE +* +* If K < 1, exit from loop +* + IF( K.LT.1 ) + $ GO TO 70 + KSTEP = 1 + P = K +* +* Determine rows and columns to be interchanged and whether +* a 1-by-1 or 2-by-2 pivot block will be used +* + ABSAKK = ABS( REAL( A( K, K ) ) ) +* +* IMAX is the row-index of the largest off-diagonal element in +* column K, and COLMAX is its absolute value. +* Determine both COLMAX and IMAX. +* + IF( K.GT.1 ) THEN + IMAX = ICAMAX( K-1, A( 1, K ), 1 ) + COLMAX = CABS1( A( IMAX, K ) ) + ELSE + COLMAX = ZERO + END IF +* + IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN +* +* Column K is zero or underflow: set INFO and continue +* + IF( INFO.EQ.0 ) + $ INFO = K + KP = K + A( K, K ) = REAL( A( K, K ) ) + ELSE +* +* ============================================================ +* +* BEGIN pivot search +* +* Case(1) +* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN +* +* no interchange, use 1-by-1 pivot block +* + KP = K +* + ELSE +* + DONE = .FALSE. +* +* Loop until pivot found +* + 12 CONTINUE +* +* BEGIN pivot search loop body +* +* +* JMAX is the column-index of the largest off-diagonal +* element in row IMAX, and ROWMAX is its absolute value. +* Determine both ROWMAX and JMAX. +* + IF( IMAX.NE.K ) THEN + JMAX = IMAX + ICAMAX( K-IMAX, A( IMAX, IMAX+1 ), + $ LDA ) + ROWMAX = CABS1( A( IMAX, JMAX ) ) + ELSE + ROWMAX = ZERO + END IF +* + IF( IMAX.GT.1 ) THEN + ITEMP = ICAMAX( IMAX-1, A( 1, IMAX ), 1 ) + STEMP = CABS1( A( ITEMP, IMAX ) ) + IF( STEMP.GT.ROWMAX ) THEN + ROWMAX = STEMP + JMAX = ITEMP + END IF + END IF +* +* Case(2) +* Equivalent to testing for +* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABS( REAL( A( IMAX, IMAX ) ) ) + $ .LT.ALPHA*ROWMAX ) ) THEN +* +* interchange rows and columns K and IMAX, +* use 1-by-1 pivot block +* + KP = IMAX + DONE = .TRUE. +* +* Case(3) +* Equivalent to testing for ROWMAX.EQ.COLMAX, +* (used to handle NaN and Inf) +* + ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) ) + $ THEN +* +* interchange rows and columns K-1 and IMAX, +* use 2-by-2 pivot block +* + KP = IMAX + KSTEP = 2 + DONE = .TRUE. +* +* Case(4) + ELSE +* +* Pivot not found: set params and repeat +* + P = IMAX + COLMAX = ROWMAX + IMAX = JMAX + END IF +* +* END pivot search loop body +* + IF( .NOT.DONE ) GOTO 12 +* + END IF +* +* END pivot search +* +* ============================================================ +* +* KK is the column of A where pivoting step stopped +* + KK = K + KSTEP - 1 +* +* For only a 2x2 pivot, interchange rows and columns K and P +* in the leading submatrix A(1:k,1:k) +* + IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN +* (1) Swap columnar parts + IF( P.GT.1 ) + $ CALL CSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 ) +* (2) Swap and conjugate middle parts + DO 14 J = P + 1, K - 1 + T = CONJG( A( J, K ) ) + A( J, K ) = CONJG( A( P, J ) ) + A( P, J ) = T + 14 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( P, K ) = CONJG( A( P, K ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = REAL( A( K, K ) ) + A( K, K ) = REAL( A( P, P ) ) + A( P, P ) = R1 + END IF +* +* For both 1x1 and 2x2 pivots, interchange rows and +* columns KK and KP in the leading submatrix A(1:k,1:k) +* + IF( KP.NE.KK ) THEN +* (1) Swap columnar parts + IF( KP.GT.1 ) + $ CALL CSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) +* (2) Swap and conjugate middle parts + DO 15 J = KP + 1, KK - 1 + T = CONJG( A( J, KK ) ) + A( J, KK ) = CONJG( A( KP, J ) ) + A( KP, J ) = T + 15 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( KP, KK ) = CONJG( A( KP, KK ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = REAL( A( KK, KK ) ) + A( KK, KK ) = REAL( A( KP, KP ) ) + A( KP, KP ) = R1 +* + IF( KSTEP.EQ.2 ) THEN +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = REAL( A( K, K ) ) +* (5) Swap row elements + T = A( K-1, K ) + A( K-1, K ) = A( KP, K ) + A( KP, K ) = T + END IF + ELSE +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = REAL( A( K, K ) ) + IF( KSTEP.EQ.2 ) + $ A( K-1, K-1 ) = REAL( A( K-1, K-1 ) ) + END IF +* +* Update the leading submatrix +* + IF( KSTEP.EQ.1 ) THEN +* +* 1-by-1 pivot block D(k): column k now holds +* +* W(k) = U(k)*D(k) +* +* where U(k) is the k-th column of U +* + IF( K.GT.1 ) THEN +* +* Perform a rank-1 update of A(1:k-1,1:k-1) and +* store U(k) in column k +* + IF( ABS( REAL( A( K, K ) ) ).GE.SFMIN ) THEN +* +* Perform a rank-1 update of A(1:k-1,1:k-1) as +* A := A - U(k)*D(k)*U(k)**T +* = A - W(k)*1/D(k)*W(k)**T +* + D11 = ONE / REAL( A( K, K ) ) + CALL CHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) +* +* Store U(k) in column k +* + CALL ZSCAL( K-1, D11, A( 1, K ), 1 ) + ELSE +* +* Store L(k) in column K +* + D11 = REAL( A( K, K ) ) + DO 16 II = 1, K - 1 + A( II, K ) = A( II, K ) / D11 + 16 CONTINUE +* +* Perform a rank-1 update of A(k+1:n,k+1:n) as +* A := A - U(k)*D(k)*U(k)**T +* = A - W(k)*(1/D(k))*W(k)**T +* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T +* + CALL CHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) + END IF + END IF +* + ELSE +* +* 2-by-2 pivot block D(k): columns k and k-1 now hold +* +* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) +* +* where U(k) and U(k-1) are the k-th and (k-1)-th columns +* of U +* +* Perform a rank-2 update of A(1:k-2,1:k-2) as +* +* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T +* = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T +* +* and store L(k) and L(k+1) in columns k and k+1 +* + IF( K.GT.2 ) THEN +* D = |A12| + D = SLAPY2( REAL( A( K-1, K ) ), + $ AIMAG( A( K-1, K ) ) ) + D11 = A( K, K ) / D + D22 = A( K-1, K-1 ) / D + D12 = A( K-1, K ) / D + TT = ONE / ( D11*D22-ONE ) +* + DO 30 J = K - 2, 1, -1 +* +* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J +* + WKM1 = TT*( D11*A( J, K-1 )-CONJG( D12 )* + $ A( J, K ) ) + WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) ) +* +* Perform a rank-2 update of A(1:k-2,1:k-2) +* + DO 20 I = J, 1, -1 + A( I, J ) = A( I, J ) - + $ ( A( I, K ) / D )*CONJG( WK ) - + $ ( A( I, K-1 ) / D )*CONJG( WKM1 ) + 20 CONTINUE +* +* Store U(k) and U(k-1) in cols k and k-1 for row J +* + A( J, K ) = WK / D + A( J, K-1 ) = WKM1 / D +* (*) Make sure that diagonal element of pivot is real + A( J, J ) = CMPLX( REAL( A( J, J ) ), ZERO ) +* + 30 CONTINUE +* + END IF +* + END IF +* + END IF +* +* Store details of the interchanges in IPIV +* + IF( KSTEP.EQ.1 ) THEN + IPIV( K ) = KP + ELSE + IPIV( K ) = -P + IPIV( K-1 ) = -KP + END IF +* +* Decrease K and return to the start of the main loop +* + K = K - KSTEP + GO TO 10 +* + ELSE +* +* Factorize A as L*D*L**H using the lower triangle of A +* +* K is the main loop index, increasing from 1 to N in steps of +* 1 or 2 +* + K = 1 + 40 CONTINUE +* +* If K > N, exit from loop +* + IF( K.GT.N ) + $ GO TO 70 + KSTEP = 1 + P = K +* +* Determine rows and columns to be interchanged and whether +* a 1-by-1 or 2-by-2 pivot block will be used +* + ABSAKK = ABS( REAL( A( K, K ) ) ) +* +* IMAX is the row-index of the largest off-diagonal element in +* column K, and COLMAX is its absolute value. +* Determine both COLMAX and IMAX. +* + IF( K.LT.N ) THEN + IMAX = K + ICAMAX( N-K, A( K+1, K ), 1 ) + COLMAX = CABS1( A( IMAX, K ) ) + ELSE + COLMAX = ZERO + END IF +* + IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN +* +* Column K is zero or underflow: set INFO and continue +* + IF( INFO.EQ.0 ) + $ INFO = K + KP = K + A( K, K ) = REAL( A( K, K ) ) + ELSE +* +* ============================================================ +* +* BEGIN pivot search +* +* Case(1) +* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN +* +* no interchange, use 1-by-1 pivot block +* + KP = K +* + ELSE +* + DONE = .FALSE. +* +* Loop until pivot found +* + 42 CONTINUE +* +* BEGIN pivot search loop body +* +* +* JMAX is the column-index of the largest off-diagonal +* element in row IMAX, and ROWMAX is its absolute value. +* Determine both ROWMAX and JMAX. +* + IF( IMAX.NE.K ) THEN + JMAX = K - 1 + ICAMAX( IMAX-K, A( IMAX, K ), LDA ) + ROWMAX = CABS1( A( IMAX, JMAX ) ) + ELSE + ROWMAX = ZERO + END IF +* + IF( IMAX.LT.N ) THEN + ITEMP = IMAX + ICAMAX( N-IMAX, A( IMAX+1, IMAX ), + $ 1 ) + STEMP = CABS1( A( ITEMP, IMAX ) ) + IF( STEMP.GT.ROWMAX ) THEN + ROWMAX = STEMP + JMAX = ITEMP + END IF + END IF +* +* Case(2) +* Equivalent to testing for +* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABS( REAL( A( IMAX, IMAX ) ) ) + $ .LT.ALPHA*ROWMAX ) ) THEN +* +* interchange rows and columns K and IMAX, +* use 1-by-1 pivot block +* + KP = IMAX + DONE = .TRUE. +* +* Case(3) +* Equivalent to testing for ROWMAX.EQ.COLMAX, +* (used to handle NaN and Inf) +* + ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) ) + $ THEN +* +* interchange rows and columns K+1 and IMAX, +* use 2-by-2 pivot block +* + KP = IMAX + KSTEP = 2 + DONE = .TRUE. +* +* Case(4) + ELSE +* +* Pivot not found: set params and repeat +* + P = IMAX + COLMAX = ROWMAX + IMAX = JMAX + END IF +* +* +* END pivot search loop body +* + IF( .NOT.DONE ) GOTO 42 +* + END IF +* +* END pivot search +* +* ============================================================ +* +* KK is the column of A where pivoting step stopped +* + KK = K + KSTEP - 1 +* +* For only a 2x2 pivot, interchange rows and columns K and P +* in the trailing submatrix A(k:n,k:n) +* + IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN +* (1) Swap columnar parts + IF( P.LT.N ) + $ CALL CSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 ) +* (2) Swap and conjugate middle parts + DO 44 J = K + 1, P - 1 + T = CONJG( A( J, K ) ) + A( J, K ) = CONJG( A( P, J ) ) + A( P, J ) = T + 44 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( P, K ) = CONJG( A( P, K ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = REAL( A( K, K ) ) + A( K, K ) = REAL( A( P, P ) ) + A( P, P ) = R1 + END IF +* +* For both 1x1 and 2x2 pivots, interchange rows and +* columns KK and KP in the trailing submatrix A(k:n,k:n) +* + IF( KP.NE.KK ) THEN +* (1) Swap columnar parts + IF( KP.LT.N ) + $ CALL CSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) +* (2) Swap and conjugate middle parts + DO 45 J = KK + 1, KP - 1 + T = CONJG( A( J, KK ) ) + A( J, KK ) = CONJG( A( KP, J ) ) + A( KP, J ) = T + 45 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( KP, KK ) = CONJG( A( KP, KK ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = REAL( A( KK, KK ) ) + A( KK, KK ) = REAL( A( KP, KP ) ) + A( KP, KP ) = R1 +* + IF( KSTEP.EQ.2 ) THEN +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = REAL( A( K, K ) ) +* (5) Swap row elements + T = A( K+1, K ) + A( K+1, K ) = A( KP, K ) + A( KP, K ) = T + END IF + ELSE +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = REAL( A( K, K ) ) + IF( KSTEP.EQ.2 ) + $ A( K+1, K+1 ) = REAL( A( K+1, K+1 ) ) + END IF +* +* Update the trailing submatrix +* + IF( KSTEP.EQ.1 ) THEN +* +* 1-by-1 pivot block D(k): column k of A now holds +* +* W(k) = L(k)*D(k), +* +* where L(k) is the k-th column of L +* + IF( K.LT.N ) THEN +* +* Perform a rank-1 update of A(k+1:n,k+1:n) and +* store L(k) in column k +* +* Handle division by a small number +* + IF( ABS( REAL( A( K, K ) ) ).GE.SFMIN ) THEN +* +* Perform a rank-1 update of A(k+1:n,k+1:n) as +* A := A - L(k)*D(k)*L(k)**T +* = A - W(k)*(1/D(k))*W(k)**T +* + D11 = ONE / REAL( A( K, K ) ) + CALL CHER( UPLO, N-K, -D11, A( K+1, K ), 1, + $ A( K+1, K+1 ), LDA ) +* +* Store L(k) in column k +* + CALL CSSCAL( N-K, D11, A( K+1, K ), 1 ) + ELSE +* +* Store L(k) in column k +* + D11 = REAL( A( K, K ) ) + DO 46 II = K + 1, N + A( II, K ) = A( II, K ) / D11 + 46 CONTINUE +* +* Perform a rank-1 update of A(k+1:n,k+1:n) as +* A := A - L(k)*D(k)*L(k)**T +* = A - W(k)*(1/D(k))*W(k)**T +* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T +* + CALL CHER( UPLO, N-K, -D11, A( K+1, K ), 1, + $ A( K+1, K+1 ), LDA ) + END IF + END IF +* + ELSE +* +* 2-by-2 pivot block D(k): columns k and k+1 now hold +* +* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) +* +* where L(k) and L(k+1) are the k-th and (k+1)-th columns +* of L +* +* +* Perform a rank-2 update of A(k+2:n,k+2:n) as +* +* A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T +* = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T +* +* and store L(k) and L(k+1) in columns k and k+1 +* + IF( K.LT.N-1 ) THEN +* D = |A21| + D = SLAPY2( REAL( A( K+1, K ) ), + $ AIMAG( A( K+1, K ) ) ) + D11 = REAL( A( K+1, K+1 ) ) / D + D22 = REAL( A( K, K ) ) / D + D21 = A( K+1, K ) / D + TT = ONE / ( D11*D22-ONE ) +* + DO 60 J = K + 2, N +* +* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J +* + WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) ) + WKP1 = TT*( D22*A( J, K+1 )-CONJG( D21 )* + $ A( J, K ) ) +* +* Perform a rank-2 update of A(k+2:n,k+2:n) +* + DO 50 I = J, N + A( I, J ) = A( I, J ) - + $ ( A( I, K ) / D )*CONJG( WK ) - + $ ( A( I, K+1 ) / D )*CONJG( WKP1 ) + 50 CONTINUE +* +* Store L(k) and L(k+1) in cols k and k+1 for row J +* + A( J, K ) = WK / D + A( J, K+1 ) = WKP1 / D +* (*) Make sure that diagonal element of pivot is real + A( J, J ) = CMPLX( REAL( A( J, J ) ), ZERO ) +* + 60 CONTINUE +* + END IF +* + END IF +* + END IF +* +* Store details of the interchanges in IPIV +* + IF( KSTEP.EQ.1 ) THEN + IPIV( K ) = KP + ELSE + IPIV( K ) = -P + IPIV( K+1 ) = -KP + END IF +* +* Increase K and return to the start of the main loop +* + K = K + KSTEP + GO TO 40 +* + END IF +* + 70 CONTINUE +* + RETURN +* +* End of CHETF2_ROOK +* + END diff --git a/SRC/zhetf2_rook.f b/SRC/zhetf2_rook.f new file mode 100644 index 00000000..ec8da924 --- /dev/null +++ b/SRC/zhetf2_rook.f @@ -0,0 +1,910 @@ +*> \brief \b ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZHETF2_ROOK + dependencies +*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetf2_rook.f"> +*> [TGZ]</a> +*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetf2_rook.f"> +*> [ZIP]</a> +*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetf2_rook.f"> +*> [TXT]</a> +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZHETF2_ROOK( UPLO, N, A, LDA, IPIV, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER UPLO +* INTEGER INFO, LDA, N +* .. +* .. Array Arguments .. +* INTEGER IPIV( * ) +* COMPLEX*16 A( LDA, * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZHETF2_ROOK computes the factorization of a complex Hermitian matrix A +*> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method: +*> +*> A = U*D*U**H or A = L*D*L**H +*> +*> where U (or L) is a product of permutation and unit upper (lower) +*> triangular matrices, U**H is the conjugate transpose of U, and D is +*> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. +*> +*> This is the unblocked version of the algorithm, calling Level 2 BLAS. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] UPLO +*> \verbatim +*> UPLO is CHARACTER*1 +*> Specifies whether the upper or lower triangular part of the +*> Hermitian matrix A is stored: +*> = 'U': Upper triangular +*> = 'L': Lower triangular +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX*16 array, dimension (LDA,N) +*> On entry, the Hermitian matrix A. If UPLO = 'U', the leading +*> n-by-n upper triangular part of A contains the upper +*> triangular part of the matrix A, and the strictly lower +*> triangular part of A is not referenced. If UPLO = 'L', the +*> leading n-by-n lower triangular part of A contains the lower +*> triangular part of the matrix A, and the strictly upper +*> triangular part of A is not referenced. +*> +*> On exit, the block diagonal matrix D and the multipliers used +*> to obtain the factor U or L (see below for further details). +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,N). +*> \endverbatim +*> +*> \param[out] IPIV +*> \verbatim +*> IPIV is INTEGER array, dimension (N) +*> Details of the interchanges and the block structure of D. +*> +*> If UPLO = 'U': +*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were +*> interchanged and D(k,k) is a 1-by-1 diagonal block. +*> +*> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and +*> columns k and -IPIV(k) were interchanged and rows and +*> columns k-1 and -IPIV(k-1) were inerchaged, +*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. +*> +*> If UPLO = 'L': +*> If IPIV(k) > 0, then rows and columns k and IPIV(k) +*> were interchanged and D(k,k) is a 1-by-1 diagonal block. +*> +*> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and +*> columns k and -IPIV(k) were interchanged and rows and +*> columns k+1 and -IPIV(k+1) were inerchaged, +*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -k, the k-th argument had an illegal value +*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization +*> has been completed, but the block diagonal matrix D is +*> exactly singular, and division by zero will occur if it +*> is used to solve a system of equations. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2012 +* +*> \ingroup complex16HEcomputational +* +*> \par Further Details: +* ===================== +*> +*> \verbatim +*> +*> If UPLO = 'U', then A = U*D*U**H, where +*> U = P(n)*U(n)* ... *P(k)U(k)* ..., +*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to +*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 +*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as +*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such +*> that if the diagonal block D(k) is of order s (s = 1 or 2), then +*> +*> ( I v 0 ) k-s +*> U(k) = ( 0 I 0 ) s +*> ( 0 0 I ) n-k +*> k-s s n-k +*> +*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). +*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), +*> and A(k,k), and v overwrites A(1:k-2,k-1:k). +*> +*> If UPLO = 'L', then A = L*D*L**H, where +*> L = P(1)*L(1)* ... *P(k)*L(k)* ..., +*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to +*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 +*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as +*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such +*> that if the diagonal block D(k) is of order s (s = 1 or 2), then +*> +*> ( I 0 0 ) k-1 +*> L(k) = ( 0 I 0 ) s +*> ( 0 v I ) n-k-s+1 +*> k-1 s n-k-s+1 +*> +*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). +*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), +*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). +*> \endverbatim +* +*> \par Contributors: +* ================== +*> +*> \verbatim +*> +*> November 2012, Igor Kozachenko, +*> Computer Science Division, +*> University of California, Berkeley +*> +*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, +*> School of Mathematics, +*> University of Manchester +*> +*> 01-01-96 - Based on modifications by +*> J. Lewis, Boeing Computer Services Company +*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA +*> \endverbatim +* +* ===================================================================== + SUBROUTINE ZHETF2_ROOK( UPLO, N, A, LDA, IPIV, INFO ) +* +* -- LAPACK computational routine (version 3.4.2) -- +* -- LAPACK is a software package provided by Univ. of Tennessee, -- +* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- +* November 2012 +* +* .. Scalar Arguments .. + CHARACTER UPLO + INTEGER INFO, LDA, N +* .. +* .. Array Arguments .. + INTEGER IPIV( * ) + COMPLEX*16 A( LDA, * ) +* .. +* +* ====================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) + DOUBLE PRECISION EIGHT, SEVTEN + PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) +* .. +* .. Local Scalars .. + LOGICAL DONE, UPPER + INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP, + $ P + DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, DTEMP, + $ ROWMAX, TT, SFMIN + COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, Z +* .. +* .. External Functions .. +* + LOGICAL LSAME + INTEGER IZAMAX + DOUBLE PRECISION DLAMCH, DLAPY2 + EXTERNAL LSAME, IZAMAX, DLAMCH, DLAPY2 +* .. +* .. External Subroutines .. + EXTERNAL XERBLA, ZDSCAL, ZHER, ZSWAP +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT +* .. +* .. Statement Functions .. + DOUBLE PRECISION CABS1 +* .. +* .. Statement Function definitions .. + CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) ) +* .. +* .. Executable Statements .. +* +* Test the input parameters. +* + INFO = 0 + UPPER = LSAME( UPLO, 'U' ) + IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -4 + END IF + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'ZHETF2_ROOK', -INFO ) + RETURN + END IF +* +* Initialize ALPHA for use in choosing pivot block size. +* + ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT +* +* Compute machine safe minimum +* + SFMIN = DLAMCH( 'S' ) +* + IF( UPPER ) THEN +* +* Factorize A as U*D*U**H using the upper triangle of A +* +* K is the main loop index, decreasing from N to 1 in steps of +* 1 or 2 +* + K = N + 10 CONTINUE +* +* If K < 1, exit from loop +* + IF( K.LT.1 ) + $ GO TO 70 + KSTEP = 1 + P = K +* +* Determine rows and columns to be interchanged and whether +* a 1-by-1 or 2-by-2 pivot block will be used +* + ABSAKK = ABS( DBLE( A( K, K ) ) ) +* +* IMAX is the row-index of the largest off-diagonal element in +* column K, and COLMAX is its absolute value. +* Determine both COLMAX and IMAX. +* + IF( K.GT.1 ) THEN + IMAX = IZAMAX( K-1, A( 1, K ), 1 ) + COLMAX = CABS1( A( IMAX, K ) ) + ELSE + COLMAX = ZERO + END IF +* + IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN +* +* Column K is zero or underflow: set INFO and continue +* + IF( INFO.EQ.0 ) + $ INFO = K + KP = K + A( K, K ) = DBLE( A( K, K ) ) + ELSE +* +* ============================================================ +* +* BEGIN pivot search +* +* Case(1) +* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN +* +* no interchange, use 1-by-1 pivot block +* + KP = K +* + ELSE +* + DONE = .FALSE. +* +* Loop until pivot found +* + 12 CONTINUE +* +* BEGIN pivot search loop body +* +* +* JMAX is the column-index of the largest off-diagonal +* element in row IMAX, and ROWMAX is its absolute value. +* Determine both ROWMAX and JMAX. +* + IF( IMAX.NE.K ) THEN + JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), + $ LDA ) + ROWMAX = CABS1( A( IMAX, JMAX ) ) + ELSE + ROWMAX = ZERO + END IF +* + IF( IMAX.GT.1 ) THEN + ITEMP = IZAMAX( IMAX-1, A( 1, IMAX ), 1 ) + DTEMP = CABS1( A( ITEMP, IMAX ) ) + IF( DTEMP.GT.ROWMAX ) THEN + ROWMAX = DTEMP + JMAX = ITEMP + END IF + END IF +* +* Case(2) +* Equivalent to testing for +* ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) ) + $ .LT.ALPHA*ROWMAX ) ) THEN +* +* interchange rows and columns K and IMAX, +* use 1-by-1 pivot block +* + KP = IMAX + DONE = .TRUE. +* +* Case(3) +* Equivalent to testing for ROWMAX.EQ.COLMAX, +* (used to handle NaN and Inf) +* + ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) ) + $ THEN +* +* interchange rows and columns K-1 and IMAX, +* use 2-by-2 pivot block +* + KP = IMAX + KSTEP = 2 + DONE = .TRUE. +* +* Case(4) + ELSE +* +* Pivot not found: set params and repeat +* + P = IMAX + COLMAX = ROWMAX + IMAX = JMAX + END IF +* +* END pivot search loop body +* + IF( .NOT.DONE ) GOTO 12 +* + END IF +* +* END pivot search +* +* ============================================================ +* +* KK is the column of A where pivoting step stopped +* + KK = K + KSTEP - 1 +* +* For only a 2x2 pivot, interchange rows and columns K and P +* in the leading submatrix A(1:k,1:k) +* + IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN +* (1) Swap columnar parts + IF( P.GT.1 ) + $ CALL ZSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 ) +* (2) Swap and conjugate middle parts + DO 14 J = P + 1, K - 1 + T = DCONJG( A( J, K ) ) + A( J, K ) = DCONJG( A( P, J ) ) + A( P, J ) = T + 14 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( P, K ) = DCONJG( A( P, K ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = DBLE( A( K, K ) ) + A( K, K ) = DBLE( A( P, P ) ) + A( P, P ) = R1 + END IF +* +* For both 1x1 and 2x2 pivots, interchange rows and +* columns KK and KP in the leading submatrix A(1:k,1:k) +* + IF( KP.NE.KK ) THEN +* (1) Swap columnar parts + IF( KP.GT.1 ) + $ CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) +* (2) Swap and conjugate middle parts + DO 15 J = KP + 1, KK - 1 + T = DCONJG( A( J, KK ) ) + A( J, KK ) = DCONJG( A( KP, J ) ) + A( KP, J ) = T + 15 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( KP, KK ) = DCONJG( A( KP, KK ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = DBLE( A( KK, KK ) ) + A( KK, KK ) = DBLE( A( KP, KP ) ) + A( KP, KP ) = R1 +* + IF( KSTEP.EQ.2 ) THEN +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = DBLE( A( K, K ) ) +* (5) Swap row elements + T = A( K-1, K ) + A( K-1, K ) = A( KP, K ) + A( KP, K ) = T + END IF + ELSE +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = DBLE( A( K, K ) ) + IF( KSTEP.EQ.2 ) + $ A( K-1, K-1 ) = DBLE( A( K-1, K-1 ) ) + END IF +* +* Update the leading submatrix +* + IF( KSTEP.EQ.1 ) THEN +* +* 1-by-1 pivot block D(k): column k now holds +* +* W(k) = U(k)*D(k) +* +* where U(k) is the k-th column of U +* + IF( K.GT.1 ) THEN +* +* Perform a rank-1 update of A(1:k-1,1:k-1) and +* store U(k) in column k +* + IF( ABS( DBLE( A( K, K ) ) ).GE.SFMIN ) THEN +* +* Perform a rank-1 update of A(1:k-1,1:k-1) as +* A := A - U(k)*D(k)*U(k)**T +* = A - W(k)*1/D(k)*W(k)**T +* + D11 = ONE / DBLE( A( K, K ) ) + CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) +* +* Store U(k) in column k +* + CALL ZSCAL( K-1, D11, A( 1, K ), 1 ) + ELSE +* +* Store L(k) in column K +* + D11 = DBLE( A( K, K ) ) + DO 16 II = 1, K - 1 + A( II, K ) = A( II, K ) / D11 + 16 CONTINUE +* +* Perform a rank-1 update of A(k+1:n,k+1:n) as +* A := A - U(k)*D(k)*U(k)**T +* = A - W(k)*(1/D(k))*W(k)**T +* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T +* + CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) + END IF + END IF +* + ELSE +* +* 2-by-2 pivot block D(k): columns k and k-1 now hold +* +* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) +* +* where U(k) and U(k-1) are the k-th and (k-1)-th columns +* of U +* +* Perform a rank-2 update of A(1:k-2,1:k-2) as +* +* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T +* = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T +* +* and store L(k) and L(k+1) in columns k and k+1 +* + IF( K.GT.2 ) THEN +* D = |A12| + D = DLAPY2( DBLE( A( K-1, K ) ), + $ DIMAG( A( K-1, K ) ) ) + D11 = A( K, K ) / D + D22 = A( K-1, K-1 ) / D + D12 = A( K-1, K ) / D + TT = ONE / ( D11*D22-ONE ) +* + DO 30 J = K - 2, 1, -1 +* +* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J +* + WKM1 = TT*( D11*A( J, K-1 )-DCONJG( D12 )* + $ A( J, K ) ) + WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) ) +* +* Perform a rank-2 update of A(1:k-2,1:k-2) +* + DO 20 I = J, 1, -1 + A( I, J ) = A( I, J ) - + $ ( A( I, K ) / D )*DCONJG( WK ) - + $ ( A( I, K-1 ) / D )*DCONJG( WKM1 ) + 20 CONTINUE +* +* Store U(k) and U(k-1) in cols k and k-1 for row J +* + A( J, K ) = WK / D + A( J, K-1 ) = WKM1 / D +* (*) Make sure that diagonal element of pivot is real + A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO ) +* + 30 CONTINUE +* + END IF +* + END IF +* + END IF +* +* Store details of the interchanges in IPIV +* + IF( KSTEP.EQ.1 ) THEN + IPIV( K ) = KP + ELSE + IPIV( K ) = -P + IPIV( K-1 ) = -KP + END IF +* +* Decrease K and return to the start of the main loop +* + K = K - KSTEP + GO TO 10 +* + ELSE +* +* Factorize A as L*D*L**H using the lower triangle of A +* +* K is the main loop index, increasing from 1 to N in steps of +* 1 or 2 +* + K = 1 + 40 CONTINUE +* +* If K > N, exit from loop +* + IF( K.GT.N ) + $ GO TO 70 + KSTEP = 1 + P = K +* +* Determine rows and columns to be interchanged and whether +* a 1-by-1 or 2-by-2 pivot block will be used +* + ABSAKK = ABS( DBLE( A( K, K ) ) ) +* +* IMAX is the row-index of the largest off-diagonal element in +* column K, and COLMAX is its absolute value. +* Determine both COLMAX and IMAX. +* + IF( K.LT.N ) THEN + IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 ) + COLMAX = CABS1( A( IMAX, K ) ) + ELSE + COLMAX = ZERO + END IF +* + IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN +* +* Column K is zero or underflow: set INFO and continue +* + IF( INFO.EQ.0 ) + $ INFO = K + KP = K + A( K, K ) = DBLE( A( K, K ) ) + ELSE +* +* ============================================================ +* +* BEGIN pivot search +* +* Case(1) +* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN +* +* no interchange, use 1-by-1 pivot block +* + KP = K +* + ELSE +* + DONE = .FALSE. +* +* Loop until pivot found +* + 42 CONTINUE +* +* BEGIN pivot search loop body +* +* +* JMAX is the column-index of the largest off-diagonal +* element in row IMAX, and ROWMAX is its absolute value. +* Determine both ROWMAX and JMAX. +* + IF( IMAX.NE.K ) THEN + JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) + ROWMAX = CABS1( A( IMAX, JMAX ) ) + ELSE + ROWMAX = ZERO + END IF +* + IF( IMAX.LT.N ) THEN + ITEMP = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), + $ 1 ) + DTEMP = CABS1( A( ITEMP, IMAX ) ) + IF( DTEMP.GT.ROWMAX ) THEN + ROWMAX = DTEMP + JMAX = ITEMP + END IF + END IF +* +* Case(2) +* Equivalent to testing for +* ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX +* (used to handle NaN and Inf) +* + IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) ) + $ .LT.ALPHA*ROWMAX ) ) THEN +* +* interchange rows and columns K and IMAX, +* use 1-by-1 pivot block +* + KP = IMAX + DONE = .TRUE. +* +* Case(3) +* Equivalent to testing for ROWMAX.EQ.COLMAX, +* (used to handle NaN and Inf) +* + ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) ) + $ THEN +* +* interchange rows and columns K+1 and IMAX, +* use 2-by-2 pivot block +* + KP = IMAX + KSTEP = 2 + DONE = .TRUE. +* +* Case(4) + ELSE +* +* Pivot not found: set params and repeat +* + P = IMAX + COLMAX = ROWMAX + IMAX = JMAX + END IF +* +* +* END pivot search loop body +* + IF( .NOT.DONE ) GOTO 42 +* + END IF +* +* END pivot search +* +* ============================================================ +* +* KK is the column of A where pivoting step stopped +* + KK = K + KSTEP - 1 +* +* For only a 2x2 pivot, interchange rows and columns K and P +* in the trailing submatrix A(k:n,k:n) +* + IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN +* (1) Swap columnar parts + IF( P.LT.N ) + $ CALL ZSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 ) +* (2) Swap and conjugate middle parts + DO 44 J = K + 1, P - 1 + T = DCONJG( A( J, K ) ) + A( J, K ) = DCONJG( A( P, J ) ) + A( P, J ) = T + 44 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( P, K ) = DCONJG( A( P, K ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = DBLE( A( K, K ) ) + A( K, K ) = DBLE( A( P, P ) ) + A( P, P ) = R1 + END IF +* +* For both 1x1 and 2x2 pivots, interchange rows and +* columns KK and KP in the trailing submatrix A(k:n,k:n) +* + IF( KP.NE.KK ) THEN +* (1) Swap columnar parts + IF( KP.LT.N ) + $ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) +* (2) Swap and conjugate middle parts + DO 45 J = KK + 1, KP - 1 + T = DCONJG( A( J, KK ) ) + A( J, KK ) = DCONJG( A( KP, J ) ) + A( KP, J ) = T + 45 CONTINUE +* (3) Swap and conjugate corner elements at row-col interserction + A( KP, KK ) = DCONJG( A( KP, KK ) ) +* (4) Swap diagonal elements at row-col intersection + R1 = DBLE( A( KK, KK ) ) + A( KK, KK ) = DBLE( A( KP, KP ) ) + A( KP, KP ) = R1 +* + IF( KSTEP.EQ.2 ) THEN +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = DBLE( A( K, K ) ) +* (5) Swap row elements + T = A( K+1, K ) + A( K+1, K ) = A( KP, K ) + A( KP, K ) = T + END IF + ELSE +* (*) Make sure that diagonal element of pivot is real + A( K, K ) = DBLE( A( K, K ) ) + IF( KSTEP.EQ.2 ) + $ A( K+1, K+1 ) = DBLE( A( K+1, K+1 ) ) + END IF +* +* Update the trailing submatrix +* + IF( KSTEP.EQ.1 ) THEN +* +* 1-by-1 pivot block D(k): column k of A now holds +* +* W(k) = L(k)*D(k), +* +* where L(k) is the k-th column of L +* + IF( K.LT.N ) THEN +* +* Perform a rank-1 update of A(k+1:n,k+1:n) and +* store L(k) in column k +* +* Handle division by a small number +* + IF( ABS( DBLE( A( K, K ) ) ).GE.SFMIN ) THEN +* +* Perform a rank-1 update of A(k+1:n,k+1:n) as +* A := A - L(k)*D(k)*L(k)**T +* = A - W(k)*(1/D(k))*W(k)**T +* + D11 = ONE / DBLE( A( K, K ) ) + CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1, + $ A( K+1, K+1 ), LDA ) +* +* Store L(k) in column k +* + CALL ZDSCAL( N-K, D11, A( K+1, K ), 1 ) + ELSE +* +* Store L(k) in column k +* + D11 = DBLE( A( K, K ) ) + DO 46 II = K + 1, N + A( II, K ) = A( II, K ) / D11 + 46 CONTINUE +* +* Perform a rank-1 update of A(k+1:n,k+1:n) as +* A := A - L(k)*D(k)*L(k)**T +* = A - W(k)*(1/D(k))*W(k)**T +* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T +* + CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1, + $ A( K+1, K+1 ), LDA ) + END IF + END IF +* + ELSE +* +* 2-by-2 pivot block D(k): columns k and k+1 now hold +* +* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) +* +* where L(k) and L(k+1) are the k-th and (k+1)-th columns +* of L +* +* +* Perform a rank-2 update of A(k+2:n,k+2:n) as +* +* A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T +* = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T +* +* and store L(k) and L(k+1) in columns k and k+1 +* + IF( K.LT.N-1 ) THEN +* D = |A21| + D = DLAPY2( DBLE( A( K+1, K ) ), + $ DIMAG( A( K+1, K ) ) ) + D11 = DBLE( A( K+1, K+1 ) ) / D + D22 = DBLE( A( K, K ) ) / D + D21 = A( K+1, K ) / D + TT = ONE / ( D11*D22-ONE ) +* + DO 60 J = K + 2, N +* +* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J +* + WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) ) + WKP1 = TT*( D22*A( J, K+1 )-DCONJG( D21 )* + $ A( J, K ) ) +* +* Perform a rank-2 update of A(k+2:n,k+2:n) +* + DO 50 I = J, N + A( I, J ) = A( I, J ) - + $ ( A( I, K ) / D )*DCONJG( WK ) - + $ ( A( I, K+1 ) / D )*DCONJG( WKP1 ) + 50 CONTINUE +* +* Store L(k) and L(k+1) in cols k and k+1 for row J +* + A( J, K ) = WK / D + A( J, K+1 ) = WKP1 / D +* (*) Make sure that diagonal element of pivot is real + A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO ) +* + 60 CONTINUE +* + END IF +* + END IF +* + END IF +* +* Store details of the interchanges in IPIV +* + IF( KSTEP.EQ.1 ) THEN + IPIV( K ) = KP + ELSE + IPIV( K ) = -P + IPIV( K+1 ) = -KP + END IF +* +* Increase K and return to the start of the main loop +* + K = K + KSTEP + GO TO 40 +* + END IF +* + 70 CONTINUE +* + RETURN +* +* End of ZHETF2_ROOK +* + END |