version 1.8, 2011/11/21 22:19:27
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version 1.14, 2016/08/27 15:27:08
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*> the right singular vectors of [A], respectively. The matrices [U] and [V] |
*> the right singular vectors of [A], respectively. The matrices [U] and [V] |
*> are computed and stored in the arrays U and V, respectively. The diagonal |
*> are computed and stored in the arrays U and V, respectively. The diagonal |
*> of [SIGMA] is computed and stored in the array SVA. |
*> of [SIGMA] is computed and stored in the array SVA. |
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*> DGEJSV can sometimes compute tiny singular values and their singular vectors much |
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*> more accurately than other SVD routines, see below under Further Details. |
*> \endverbatim |
*> \endverbatim |
* |
* |
* Arguments: |
* Arguments: |
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*> copied back to the V array. This 'W' option is just |
*> copied back to the V array. This 'W' option is just |
*> a reminder to the caller that in this case U is |
*> a reminder to the caller that in this case U is |
*> reserved as workspace of length N*N. |
*> reserved as workspace of length N*N. |
*> If JOBU = 'N' U is not referenced. |
*> If JOBU = 'N' U is not referenced, unless JOBT='T'. |
*> \endverbatim |
*> \endverbatim |
*> |
*> |
*> \param[in] LDU |
*> \param[in] LDU |
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*> copied back to the U array. This 'W' option is just |
*> copied back to the U array. This 'W' option is just |
*> a reminder to the caller that in this case V is |
*> a reminder to the caller that in this case V is |
*> reserved as workspace of length N*N. |
*> reserved as workspace of length N*N. |
*> If JOBV = 'N' V is not referenced. |
*> If JOBV = 'N' V is not referenced, unless JOBT='T'. |
*> \endverbatim |
*> \endverbatim |
*> |
*> |
*> \param[in] LDV |
*> \param[in] LDV |
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*> If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'), |
*> If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'), |
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). |
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). |
*> -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7), |
*> -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7), |
*> where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ, |
*> where NB is the optimal block size for DGEQP3, DGEQRF, DGELQF, |
*> DORMLQ. In general, the optimal length LWORK is computed as |
*> DORMLQ. In general, the optimal length LWORK is computed as |
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), |
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), |
*> N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)). |
*> N+LWORK(DGELQF), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)). |
*> |
*> |
*> If SIGMA and the left singular vectors are needed |
*> If SIGMA and the left singular vectors are needed |
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). |
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). |
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*> \author Univ. of Colorado Denver |
*> \author Univ. of Colorado Denver |
*> \author NAG Ltd. |
*> \author NAG Ltd. |
* |
* |
*> \date November 2011 |
*> \date June 2016 |
* |
* |
*> \ingroup doubleGEcomputational |
*> \ingroup doubleGEsing |
* |
* |
*> \par Further Details: |
*> \par Further Details: |
* ===================== |
* ===================== |
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$ M, N, A, LDA, SVA, U, LDU, V, LDV, |
$ M, N, A, LDA, SVA, U, LDU, V, LDV, |
$ WORK, LWORK, IWORK, INFO ) |
$ WORK, LWORK, IWORK, INFO ) |
* |
* |
* -- LAPACK computational routine (version 3.4.0) -- |
* -- LAPACK computational routine (version 3.6.1) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2011 |
* June 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
IMPLICIT NONE |
IMPLICIT NONE |
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$ NOSCAL, ROWPIV, RSVEC, TRANSP |
$ NOSCAL, ROWPIV, RSVEC, TRANSP |
* .. |
* .. |
* .. Intrinsic Functions .. |
* .. Intrinsic Functions .. |
INTRINSIC DABS, DLOG, DMAX1, DMIN1, DBLE, |
INTRINSIC DABS, DLOG, MAX, MIN, DBLE, IDNINT, DSIGN, DSQRT |
$ MAX0, MIN0, IDNINT, DSIGN, DSQRT |
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* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
DOUBLE PRECISION DLAMCH, DNRM2 |
DOUBLE PRECISION DLAMCH, DNRM2 |
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ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN |
ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN |
INFO = - 14 |
INFO = - 14 |
ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND. |
ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND. |
& (LWORK .LT. MAX0(7,4*N+1,2*M+N))) .OR. |
& (LWORK .LT. MAX(7,4*N+1,2*M+N))) .OR. |
& (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND. |
& (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND. |
& (LWORK .LT. MAX0(7,4*N+N*N,2*M+N))) .OR. |
& (LWORK .LT. MAX(7,4*N+N*N,2*M+N))) .OR. |
& (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX0(7,2*M+N,4*N+1))) |
& (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1))) |
& .OR. |
& .OR. |
& (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX0(7,2*M+N,4*N+1))) |
& (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1))) |
& .OR. |
& .OR. |
& (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND. |
& (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND. |
& (LWORK.LT.MAX0(2*M+N,6*N+2*N*N))) |
& (LWORK.LT.MAX(2*M+N,6*N+2*N*N))) |
& .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND. |
& .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND. |
& LWORK.LT.MAX0(2*M+N,4*N+N*N,2*N+N*N+6))) |
& LWORK.LT.MAX(2*M+N,4*N+N*N,2*N+N*N+6))) |
& THEN |
& THEN |
INFO = - 17 |
INFO = - 17 |
ELSE |
ELSE |
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* |
* |
* Quick return for void matrix (Y3K safe) |
* Quick return for void matrix (Y3K safe) |
* #:) |
* #:) |
IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) RETURN |
IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN |
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IWORK(1:3) = 0 |
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WORK(1:7) = 0 |
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RETURN |
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ENDIF |
* |
* |
* Determine whether the matrix U should be M x N or M x M |
* Determine whether the matrix U should be M x N or M x M |
* |
* |
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AAPP = ZERO |
AAPP = ZERO |
AAQQ = BIG |
AAQQ = BIG |
DO 4781 p = 1, N |
DO 4781 p = 1, N |
AAPP = DMAX1( AAPP, SVA(p) ) |
AAPP = MAX( AAPP, SVA(p) ) |
IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) ) |
IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) ) |
4781 CONTINUE |
4781 CONTINUE |
* |
* |
* Quick return for zero M x N matrix |
* Quick return for zero M x N matrix |
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IWORK(1) = 0 |
IWORK(1) = 0 |
IWORK(2) = 0 |
IWORK(2) = 0 |
END IF |
END IF |
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IWORK(3) = 0 |
IF ( ERREST ) WORK(3) = ONE |
IF ( ERREST ) WORK(3) = ONE |
IF ( LSVEC .AND. RSVEC ) THEN |
IF ( LSVEC .AND. RSVEC ) THEN |
WORK(4) = ONE |
WORK(4) = ONE |
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* in one pass through the vector |
* in one pass through the vector |
WORK(M+N+p) = XSC * SCALEM |
WORK(M+N+p) = XSC * SCALEM |
WORK(N+p) = XSC * (SCALEM*DSQRT(TEMP1)) |
WORK(N+p) = XSC * (SCALEM*DSQRT(TEMP1)) |
AATMAX = DMAX1( AATMAX, WORK(N+p) ) |
AATMAX = MAX( AATMAX, WORK(N+p) ) |
IF (WORK(N+p) .NE. ZERO) AATMIN = DMIN1(AATMIN,WORK(N+p)) |
IF (WORK(N+p) .NE. ZERO) AATMIN = MIN(AATMIN,WORK(N+p)) |
1950 CONTINUE |
1950 CONTINUE |
ELSE |
ELSE |
DO 1904 p = 1, M |
DO 1904 p = 1, M |
WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) ) |
WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) ) |
AATMAX = DMAX1( AATMAX, WORK(M+N+p) ) |
AATMAX = MAX( AATMAX, WORK(M+N+p) ) |
AATMIN = DMIN1( AATMIN, WORK(M+N+p) ) |
AATMIN = MIN( AATMIN, WORK(M+N+p) ) |
1904 CONTINUE |
1904 CONTINUE |
END IF |
END IF |
* |
* |
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MAXPRJ = ONE |
MAXPRJ = ONE |
DO 3051 p = 2, N |
DO 3051 p = 2, N |
TEMP1 = DABS(A(p,p)) / SVA(IWORK(p)) |
TEMP1 = DABS(A(p,p)) / SVA(IWORK(p)) |
MAXPRJ = DMIN1( MAXPRJ, TEMP1 ) |
MAXPRJ = MIN( MAXPRJ, TEMP1 ) |
3051 CONTINUE |
3051 CONTINUE |
IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE. |
IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE. |
END IF |
END IF |
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* Singular Values only |
* Singular Values only |
* |
* |
* .. transpose A(1:NR,1:N) |
* .. transpose A(1:NR,1:N) |
DO 1946 p = 1, MIN0( N-1, NR ) |
DO 1946 p = 1, MIN( N-1, NR ) |
CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 ) |
CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 ) |
1946 CONTINUE |
1946 CONTINUE |
* |
* |
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XSC = DSQRT(SMALL)/EPSLN |
XSC = DSQRT(SMALL)/EPSLN |
DO 3959 p = 2, NR |
DO 3959 p = 2, NR |
DO 3958 q = 1, p - 1 |
DO 3958 q = 1, p - 1 |
TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q))) |
TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q))) |
IF ( DABS(V(q,p)) .LE. TEMP1 ) |
IF ( DABS(V(q,p)) .LE. TEMP1 ) |
$ V(q,p) = DSIGN( TEMP1, V(q,p) ) |
$ V(q,p) = DSIGN( TEMP1, V(q,p) ) |
3958 CONTINUE |
3958 CONTINUE |
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XSC = DSQRT(SMALL) |
XSC = DSQRT(SMALL) |
DO 3969 p = 2, NR |
DO 3969 p = 2, NR |
DO 3968 q = 1, p - 1 |
DO 3968 q = 1, p - 1 |
TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q))) |
TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q))) |
IF ( DABS(V(q,p)) .LE. TEMP1 ) |
IF ( DABS(V(q,p)) .LE. TEMP1 ) |
$ V(q,p) = DSIGN( TEMP1, V(q,p) ) |
$ V(q,p) = DSIGN( TEMP1, V(q,p) ) |
3968 CONTINUE |
3968 CONTINUE |
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XSC = DSQRT(SMALL) |
XSC = DSQRT(SMALL) |
DO 8970 p = 2, NR |
DO 8970 p = 2, NR |
DO 8971 q = 1, p - 1 |
DO 8971 q = 1, p - 1 |
TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q))) |
TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q))) |
V(p,q) = - DSIGN( TEMP1, V(q,p) ) |
V(p,q) = - DSIGN( TEMP1, V(q,p) ) |
8971 CONTINUE |
8971 CONTINUE |
8970 CONTINUE |
8970 CONTINUE |
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XSC = DSQRT(SMALL/EPSLN) |
XSC = DSQRT(SMALL/EPSLN) |
DO 9970 q = 2, NR |
DO 9970 q = 2, NR |
DO 9971 p = 1, q - 1 |
DO 9971 p = 1, q - 1 |
TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q))) |
TEMP1 = XSC * MIN(DABS(U(p,p)),DABS(U(q,q))) |
U(p,q) = - DSIGN( TEMP1, U(q,p) ) |
U(p,q) = - DSIGN( TEMP1, U(q,p) ) |
9971 CONTINUE |
9971 CONTINUE |
9970 CONTINUE |
9970 CONTINUE |