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*> \brief \b CTGSJA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CTGSJA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctgsja.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ctgsja.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ctgsja.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CTGSJA( 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 )
*
* .. Scalar Arguments ..
* CHARACTER JOBQ, JOBU, JOBV
* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
* $ NCYCLE, P
* REAL TOLA, TOLB
* ..
* .. Array Arguments ..
* REAL ALPHA( * ), BETA( * )
* COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CTGSJA computes the generalized singular value decomposition (GSVD)
*> of two complex 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 CGGSVP
*> 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**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
*>
*> where U, V and Q are unitary matrices.
*> 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 unitary 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'U': U must contain a unitary matrix U1 on entry, and
*> the product U1*U is returned;
*> = 'I': U is initialized to the unit matrix, and the
*> unitary matrix U is returned;
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'V': V must contain a unitary matrix V1 on entry, and
*> the product V1*V is returned;
*> = 'I': V is initialized to the unit matrix, and the
*> unitary matrix V is returned;
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*> JOBQ is CHARACTER*1
*> = 'Q': Q must contain a unitary matrix Q1 on entry, and
*> the product Q1*Q is returned;
*> = 'I': Q is initialized to the unit matrix, and the
*> unitary matrix Q is returned;
*> = 'N': Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is 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 CTGSJA.
*> See Further Details.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX 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.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in] TOLA
*> \verbatim
*> TOLA is REAL
*> \endverbatim
*>
*> \param[in] TOLB
*> \verbatim
*> TOLB is REAL
*>
*> 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)*MACHEPS,
*> TOLB = MAX(P,N)*norm(B)*MACHEPS.
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is REAL 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
*> BETA(K+L+1:N) = 0.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is COMPLEX array, dimension (LDU,M)
*> On entry, if JOBU = 'U', U must contain a matrix U1 (usually
*> the unitary matrix returned by CGGSVP).
*> On exit,
*> if JOBU = 'I', U contains the unitary matrix U;
*> if JOBU = 'U', U contains the product U1*U.
*> If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,M) if
*> JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is COMPLEX array, dimension (LDV,P)
*> On entry, if JOBV = 'V', V must contain a matrix V1 (usually
*> the unitary matrix returned by CGGSVP).
*> On exit,
*> if JOBV = 'I', V contains the unitary matrix V;
*> if JOBV = 'V', V contains the product V1*V.
*> If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,P) if
*> JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is COMPLEX array, dimension (LDQ,N)
*> On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
*> the unitary matrix returned by CGGSVP).
*> On exit,
*> if JOBQ = 'I', Q contains the unitary matrix Q;
*> if JOBQ = 'Q', Q contains the product Q1*Q.
*> If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N) if
*> JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] NCYCLE
*> \verbatim
*> NCYCLE is INTEGER
*> The number of cycles required for convergence.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is 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.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complexOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> CTGSJA 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**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
*>
*> where U1, V1 and Q1 are unitary matrix.
*> C1 and S1 are diagonal matrices satisfying
*>
*> C1**2 + S1**2 = I,
*>
*> and R1 is an L-by-L nonsingular upper triangular matrix.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CTGSJA( 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 computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
$ NCYCLE, P
REAL TOLA, TOLB
* ..
* .. Array Arguments ..
REAL ALPHA( * ), BETA( * )
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 40 )
REAL ZERO, ONE, HUGENUM
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
*
LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
INTEGER I, J, KCYCLE
REAL A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA,
$ RWK, SSMIN
COMPLEX A2, B2, SNQ, SNU, SNV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CLAGS2, CLAPLL, CLASET, CROT, CSSCAL,
$ SLARTG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, MIN, REAL, HUGE
PARAMETER ( HUGENUM = HUGE(ZERO) )
* ..
* .. 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( 'CTGSJA', -INFO )
RETURN
END IF
*
* Initialize U, V and Q, if necessary
*
IF( INITU )
$ CALL CLASET( 'Full', M, M, CZERO, CONE, U, LDU )
IF( INITV )
$ CALL CLASET( 'Full', P, P, CZERO, CONE, V, LDV )
IF( INITQ )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, 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 = CZERO
A3 = ZERO
IF( K+I.LE.M )
$ A1 = REAL( A( K+I, N-L+I ) )
IF( K+J.LE.M )
$ A3 = REAL( A( K+J, N-L+J ) )
*
B1 = REAL( B( I, N-L+I ) )
B3 = REAL( 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 CLAGS2( 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**H *A
*
IF( K+J.LE.M )
$ CALL CROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
$ LDA, CSU, CONJG( SNU ) )
*
* Update I-th and J-th rows of matrix B: V**H *B
*
CALL CROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
$ CSV, CONJG( SNV ) )
*
* Update (N-L+I)-th and (N-L+J)-th columns of matrices
* A and B: A*Q and B*Q
*
CALL CROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
$ A( 1, N-L+I ), 1, CSQ, SNQ )
*
CALL CROT( 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 ) = CZERO
B( I, N-L+J ) = CZERO
ELSE
IF( K+J.LE.M )
$ A( K+J, N-L+I ) = CZERO
B( J, N-L+I ) = CZERO
END IF
*
* Ensure that the diagonal elements of A and B are real.
*
IF( K+I.LE.M )
$ A( K+I, N-L+I ) = REAL( A( K+I, N-L+I ) )
IF( K+J.LE.M )
$ A( K+J, N-L+J ) = REAL( A( K+J, N-L+J ) )
B( I, N-L+I ) = REAL( B( I, N-L+I ) )
B( J, N-L+J ) = REAL( B( J, N-L+J ) )
*
* Update unitary matrices U, V, Q, if desired.
*
IF( WANTU .AND. K+J.LE.M )
$ CALL CROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
$ SNU )
*
IF( WANTV )
$ CALL CROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
*
IF( WANTQ )
$ CALL CROT( 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 CCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
CALL CCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
CALL CLAPLL( 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 = REAL( A( K+I, N-L+I ) )
B1 = REAL( B( I, N-L+I ) )
GAMMA = B1 / A1
*
IF( (GAMMA.LE.HUGENUM).AND.(GAMMA.GE.-HUGENUM) ) THEN
*
IF( GAMMA.LT.ZERO ) THEN
CALL CSSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
IF( WANTV )
$ CALL CSSCAL( P, -ONE, V( 1, I ), 1 )
END IF
*
CALL SLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
$ RWK )
*
IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
CALL CSSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
$ LDA )
ELSE
CALL CSSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
$ LDB )
CALL CCOPY( 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 CCOPY( 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 CTGSJA
*
END
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