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authorjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
committerjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
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+ SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
+ $ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
+ $ Q, LDQ, WORK, NCYCLE, INFO )
+*
+* -- LAPACK routine (version 3.1) --
+* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+* November 2006
+*
+* .. Scalar Arguments ..
+ CHARACTER JOBQ, JOBU, JOBV
+ INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
+ $ NCYCLE, P
+ DOUBLE PRECISION TOLA, TOLB
+* ..
+* .. Array Arguments ..
+ DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
+ $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
+ $ V( LDV, * ), WORK( * )
+* ..
+*
+* Purpose
+* =======
+*
+* DTGSJA computes the generalized singular value decomposition (GSVD)
+* of two real upper triangular (or trapezoidal) matrices A and B.
+*
+* On entry, it is assumed that matrices A and B have the following
+* forms, which may be obtained by the preprocessing subroutine DGGSVP
+* from a general M-by-N matrix A and P-by-N matrix B:
+*
+* N-K-L K L
+* A = K ( 0 A12 A13 ) if M-K-L >= 0;
+* L ( 0 0 A23 )
+* M-K-L ( 0 0 0 )
+*
+* N-K-L K L
+* A = K ( 0 A12 A13 ) if M-K-L < 0;
+* M-K ( 0 0 A23 )
+*
+* N-K-L K L
+* B = L ( 0 0 B13 )
+* P-L ( 0 0 0 )
+*
+* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
+* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
+* otherwise A23 is (M-K)-by-L upper trapezoidal.
+*
+* On exit,
+*
+* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
+*
+* where U, V and Q are orthogonal matrices, Z' denotes the transpose
+* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
+* ``diagonal'' matrices, which are of the following structures:
+*
+* If M-K-L >= 0,
+*
+* K L
+* D1 = K ( I 0 )
+* L ( 0 C )
+* M-K-L ( 0 0 )
+*
+* K L
+* D2 = L ( 0 S )
+* P-L ( 0 0 )
+*
+* N-K-L K L
+* ( 0 R ) = K ( 0 R11 R12 ) K
+* L ( 0 0 R22 ) L
+*
+* where
+*
+* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
+* S = diag( BETA(K+1), ... , BETA(K+L) ),
+* C**2 + S**2 = I.
+*
+* R is stored in A(1:K+L,N-K-L+1:N) on exit.
+*
+* If M-K-L < 0,
+*
+* K M-K K+L-M
+* D1 = K ( I 0 0 )
+* M-K ( 0 C 0 )
+*
+* K M-K K+L-M
+* D2 = M-K ( 0 S 0 )
+* K+L-M ( 0 0 I )
+* P-L ( 0 0 0 )
+*
+* N-K-L K M-K K+L-M
+* ( 0 R ) = K ( 0 R11 R12 R13 )
+* M-K ( 0 0 R22 R23 )
+* K+L-M ( 0 0 0 R33 )
+*
+* where
+* C = diag( ALPHA(K+1), ... , ALPHA(M) ),
+* S = diag( BETA(K+1), ... , BETA(M) ),
+* C**2 + S**2 = I.
+*
+* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
+* ( 0 R22 R23 )
+* in B(M-K+1:L,N+M-K-L+1:N) on exit.
+*
+* The computation of the orthogonal transformation matrices U, V or Q
+* is optional. These matrices may either be formed explicitly, or they
+* may be postmultiplied into input matrices U1, V1, or Q1.
+*
+* Arguments
+* =========
+*
+* JOBU (input) CHARACTER*1
+* = 'U': U must contain an orthogonal matrix U1 on entry, and
+* the product U1*U is returned;
+* = 'I': U is initialized to the unit matrix, and the
+* orthogonal matrix U is returned;
+* = 'N': U is not computed.
+*
+* JOBV (input) CHARACTER*1
+* = 'V': V must contain an orthogonal matrix V1 on entry, and
+* the product V1*V is returned;
+* = 'I': V is initialized to the unit matrix, and the
+* orthogonal matrix V is returned;
+* = 'N': V is not computed.
+*
+* JOBQ (input) CHARACTER*1
+* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
+* the product Q1*Q is returned;
+* = 'I': Q is initialized to the unit matrix, and the
+* orthogonal matrix Q is returned;
+* = 'N': Q is not computed.
+*
+* M (input) INTEGER
+* The number of rows of the matrix A. M >= 0.
+*
+* P (input) INTEGER
+* The number of rows of the matrix B. P >= 0.
+*
+* N (input) INTEGER
+* The number of columns of the matrices A and B. N >= 0.
+*
+* K (input) INTEGER
+* L (input) INTEGER
+* K and L specify the subblocks in the input matrices A and B:
+* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
+* of A and B, whose GSVD is going to be computed by DTGSJA.
+* See Further details.
+*
+* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
+* On entry, the M-by-N matrix A.
+* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
+* matrix R or part of R. See Purpose for details.
+*
+* LDA (input) INTEGER
+* The leading dimension of the array A. LDA >= max(1,M).
+*
+* B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
+* On entry, the P-by-N matrix B.
+* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
+* a part of R. See Purpose for details.
+*
+* LDB (input) INTEGER
+* The leading dimension of the array B. LDB >= max(1,P).
+*
+* TOLA (input) DOUBLE PRECISION
+* TOLB (input) DOUBLE PRECISION
+* TOLA and TOLB are the convergence criteria for the Jacobi-
+* Kogbetliantz iteration procedure. Generally, they are the
+* same as used in the preprocessing step, say
+* TOLA = max(M,N)*norm(A)*MAZHEPS,
+* TOLB = max(P,N)*norm(B)*MAZHEPS.
+*
+* ALPHA (output) DOUBLE PRECISION array, dimension (N)
+* BETA (output) DOUBLE PRECISION array, dimension (N)
+* On exit, ALPHA and BETA contain the generalized singular
+* value pairs of A and B;
+* ALPHA(1:K) = 1,
+* BETA(1:K) = 0,
+* and if M-K-L >= 0,
+* ALPHA(K+1:K+L) = diag(C),
+* BETA(K+1:K+L) = diag(S),
+* or if M-K-L < 0,
+* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
+* BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
+* Furthermore, if K+L < N,
+* ALPHA(K+L+1:N) = 0 and
+* BETA(K+L+1:N) = 0.
+*
+* U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
+* On entry, if JOBU = 'U', U must contain a matrix U1 (usually
+* the orthogonal matrix returned by DGGSVP).
+* On exit,
+* if JOBU = 'I', U contains the orthogonal matrix U;
+* if JOBU = 'U', U contains the product U1*U.
+* If JOBU = 'N', U is not referenced.
+*
+* LDU (input) INTEGER
+* The leading dimension of the array U. LDU >= max(1,M) if
+* JOBU = 'U'; LDU >= 1 otherwise.
+*
+* V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
+* On entry, if JOBV = 'V', V must contain a matrix V1 (usually
+* the orthogonal matrix returned by DGGSVP).
+* On exit,
+* if JOBV = 'I', V contains the orthogonal matrix V;
+* if JOBV = 'V', V contains the product V1*V.
+* If JOBV = 'N', V is not referenced.
+*
+* LDV (input) INTEGER
+* The leading dimension of the array V. LDV >= max(1,P) if
+* JOBV = 'V'; LDV >= 1 otherwise.
+*
+* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
+* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
+* the orthogonal matrix returned by DGGSVP).
+* On exit,
+* if JOBQ = 'I', Q contains the orthogonal matrix Q;
+* if JOBQ = 'Q', Q contains the product Q1*Q.
+* If JOBQ = 'N', Q is not referenced.
+*
+* LDQ (input) INTEGER
+* The leading dimension of the array Q. LDQ >= max(1,N) if
+* JOBQ = 'Q'; LDQ >= 1 otherwise.
+*
+* WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
+*
+* NCYCLE (output) INTEGER
+* The number of cycles required for convergence.
+*
+* INFO (output) INTEGER
+* = 0: successful exit
+* < 0: if INFO = -i, the i-th argument had an illegal value.
+* = 1: the procedure does not converge after MAXIT cycles.
+*
+* Internal Parameters
+* ===================
+*
+* MAXIT INTEGER
+* MAXIT specifies the total loops that the iterative procedure
+* may take. If after MAXIT cycles, the routine fails to
+* converge, we return INFO = 1.
+*
+* Further Details
+* ===============
+*
+* DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
+* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
+* matrix B13 to the form:
+*
+* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
+*
+* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
+* of Z. C1 and S1 are diagonal matrices satisfying
+*
+* C1**2 + S1**2 = I,
+*
+* and R1 is an L-by-L nonsingular upper triangular matrix.
+*
+* =====================================================================
+*
+* .. Parameters ..
+ INTEGER MAXIT
+ PARAMETER ( MAXIT = 40 )
+ DOUBLE PRECISION ZERO, ONE
+ PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
+* ..
+* .. Local Scalars ..
+*
+ LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
+ INTEGER I, J, KCYCLE
+ DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
+ $ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
+* ..
+* .. External Functions ..
+ LOGICAL LSAME
+ EXTERNAL LSAME
+* ..
+* .. External Subroutines ..
+ EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT,
+ $ DSCAL, XERBLA
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, MAX, MIN
+* ..
+* .. Executable Statements ..
+*
+* Decode and test the input parameters
+*
+ INITU = LSAME( JOBU, 'I' )
+ WANTU = INITU .OR. LSAME( JOBU, 'U' )
+*
+ INITV = LSAME( JOBV, 'I' )
+ WANTV = INITV .OR. LSAME( JOBV, 'V' )
+*
+ INITQ = LSAME( JOBQ, 'I' )
+ WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
+*
+ INFO = 0
+ IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
+ INFO = -1
+ ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
+ INFO = -2
+ ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
+ INFO = -3
+ ELSE IF( M.LT.0 ) THEN
+ INFO = -4
+ ELSE IF( P.LT.0 ) THEN
+ INFO = -5
+ ELSE IF( N.LT.0 ) THEN
+ INFO = -6
+ ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
+ INFO = -10
+ ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
+ INFO = -12
+ ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
+ INFO = -18
+ ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
+ INFO = -20
+ ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
+ INFO = -22
+ END IF
+ IF( INFO.NE.0 ) THEN
+ CALL XERBLA( 'DTGSJA', -INFO )
+ RETURN
+ END IF
+*
+* Initialize U, V and Q, if necessary
+*
+ IF( INITU )
+ $ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
+ IF( INITV )
+ $ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
+ IF( INITQ )
+ $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
+*
+* Loop until convergence
+*
+ UPPER = .FALSE.
+ DO 40 KCYCLE = 1, MAXIT
+*
+ UPPER = .NOT.UPPER
+*
+ DO 20 I = 1, L - 1
+ DO 10 J = I + 1, L
+*
+ A1 = ZERO
+ A2 = ZERO
+ A3 = ZERO
+ IF( K+I.LE.M )
+ $ A1 = A( K+I, N-L+I )
+ IF( K+J.LE.M )
+ $ A3 = A( K+J, N-L+J )
+*
+ B1 = B( I, N-L+I )
+ B3 = B( J, N-L+J )
+*
+ IF( UPPER ) THEN
+ IF( K+I.LE.M )
+ $ A2 = A( K+I, N-L+J )
+ B2 = B( I, N-L+J )
+ ELSE
+ IF( K+J.LE.M )
+ $ A2 = A( K+J, N-L+I )
+ B2 = B( J, N-L+I )
+ END IF
+*
+ CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
+ $ CSV, SNV, CSQ, SNQ )
+*
+* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A
+*
+ IF( K+J.LE.M )
+ $ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
+ $ LDA, CSU, SNU )
+*
+* Update I-th and J-th rows of matrix B: V'*B
+*
+ CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
+ $ CSV, SNV )
+*
+* Update (N-L+I)-th and (N-L+J)-th columns of matrices
+* A and B: A*Q and B*Q
+*
+ CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
+ $ A( 1, N-L+I ), 1, CSQ, SNQ )
+*
+ CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
+ $ SNQ )
+*
+ IF( UPPER ) THEN
+ IF( K+I.LE.M )
+ $ A( K+I, N-L+J ) = ZERO
+ B( I, N-L+J ) = ZERO
+ ELSE
+ IF( K+J.LE.M )
+ $ A( K+J, N-L+I ) = ZERO
+ B( J, N-L+I ) = ZERO
+ END IF
+*
+* Update orthogonal matrices U, V, Q, if desired.
+*
+ IF( WANTU .AND. K+J.LE.M )
+ $ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
+ $ SNU )
+*
+ IF( WANTV )
+ $ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
+*
+ IF( WANTQ )
+ $ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
+ $ SNQ )
+*
+ 10 CONTINUE
+ 20 CONTINUE
+*
+ IF( .NOT.UPPER ) THEN
+*
+* The matrices A13 and B13 were lower triangular at the start
+* of the cycle, and are now upper triangular.
+*
+* Convergence test: test the parallelism of the corresponding
+* rows of A and B.
+*
+ ERROR = ZERO
+ DO 30 I = 1, MIN( L, M-K )
+ CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
+ CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
+ CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
+ ERROR = MAX( ERROR, SSMIN )
+ 30 CONTINUE
+*
+ IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
+ $ GO TO 50
+ END IF
+*
+* End of cycle loop
+*
+ 40 CONTINUE
+*
+* The algorithm has not converged after MAXIT cycles.
+*
+ INFO = 1
+ GO TO 100
+*
+ 50 CONTINUE
+*
+* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
+* Compute the generalized singular value pairs (ALPHA, BETA), and
+* set the triangular matrix R to array A.
+*
+ DO 60 I = 1, K
+ ALPHA( I ) = ONE
+ BETA( I ) = ZERO
+ 60 CONTINUE
+*
+ DO 70 I = 1, MIN( L, M-K )
+*
+ A1 = A( K+I, N-L+I )
+ B1 = B( I, N-L+I )
+*
+ IF( A1.NE.ZERO ) THEN
+ GAMMA = B1 / A1
+*
+* change sign if necessary
+*
+ IF( GAMMA.LT.ZERO ) THEN
+ CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
+ IF( WANTV )
+ $ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
+ END IF
+*
+ CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
+ $ RWK )
+*
+ IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
+ CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
+ $ LDA )
+ ELSE
+ CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
+ $ LDB )
+ CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
+ $ LDA )
+ END IF
+*
+ ELSE
+*
+ ALPHA( K+I ) = ZERO
+ BETA( K+I ) = ONE
+ CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
+ $ LDA )
+*
+ END IF
+*
+ 70 CONTINUE
+*
+* Post-assignment
+*
+ DO 80 I = M + 1, K + L
+ ALPHA( I ) = ZERO
+ BETA( I ) = ONE
+ 80 CONTINUE
+*
+ IF( K+L.LT.N ) THEN
+ DO 90 I = K + L + 1, N
+ ALPHA( I ) = ZERO
+ BETA( I ) = ZERO
+ 90 CONTINUE
+ END IF
+*
+ 100 CONTINUE
+ NCYCLE = KCYCLE
+ RETURN
+*
+* End of DTGSJA
+*
+ END