gesdd(3)

NAME
gesdd - gesdd: SVD, divide and conquer

SYNOPSIS

  Functions
subroutine cgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)
CGESDD
subroutine dgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, info)
DGESDD
subroutine sgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, info)
SGESDD
subroutine zgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)
ZGESDD

Detailed Description

Function Documentation

  subroutine cgesdd (character jobz, integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) s, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldvt, * ) vt, integer ldvt, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CGESDD

Purpose:

     CGESDD computes the singular value decomposition (SVD) of a complex
     M-by-N matrix A, optionally computing the left and/or right singular
     vectors, by using divide-and-conquer method. The SVD is written
          A = U * SIGMA * conjugate-transpose(V)
     where SIGMA is an M-by-N matrix which is zero except for its
     min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
     V is an N-by-N unitary matrix.  The diagonal elements of SIGMA
     are the singular values of A; they are real and non-negative, and
     are returned in descending order.  The first min(m,n) columns of
     U and V are the left and right singular vectors of A.
     Note that the routine returns VT = V**H, not V.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'A':  all M columns of U and all N rows of V**H are
                      returned in the arrays U and VT;
              = 'S':  the first min(M,N) columns of U and the first
                      min(M,N) rows of V**H are returned in the arrays U
                      and VT;
              = 'O':  If M >= N, the first N columns of U are overwritten
                      in the array A and all rows of V**H are returned in
                      the array VT;
                      otherwise, all columns of U are returned in the
                      array U and the first M rows of V**H are overwritten
                      in the array A;
              = 'N':  no columns of U or rows of V**H are computed.

    M

              M is INTEGER
              The number of rows of the input matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the input matrix A.  N >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if JOBZ = 'O',  A is overwritten with the first N columns
                              of U (the left singular vectors, stored
                              columnwise) if M >= N;
                              A is overwritten with the first M rows
                              of V**H (the right singular vectors, stored
                              rowwise) otherwise.
              if JOBZ .ne. 'O', the contents of A are destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    S

              S is REAL array, dimension (min(M,N))
              The singular values of A, sorted so that S(i) >= S(i+1).

    U

              U is COMPLEX array, dimension (LDU,UCOL)
              UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
              UCOL = min(M,N) if JOBZ = 'S'.
              If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
              unitary matrix U;
              if JOBZ = 'S', U contains the first min(M,N) columns of U
              (the left singular vectors, stored columnwise);
              if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1;
              if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

    VT

              VT is COMPLEX array, dimension (LDVT,N)
              If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
              N-by-N unitary matrix V**H;
              if JOBZ = 'S', VT contains the first min(M,N) rows of
              V**H (the right singular vectors, stored rowwise);
              if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1;
              if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
              if JOBZ = 'S', LDVT >= min(M,N).

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= 1.
              If LWORK = -1, a workspace query is assumed.  The optimal
              size for the WORK array is calculated and stored in WORK(1),
              and no other work except argument checking is performed.
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N', LWORK >= 2*mn + mx.
              If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
              If JOBZ = 'S', LWORK >=   mn*mn + 3*mn.
              If JOBZ = 'A', LWORK >=   mn*mn + 2*mn + mx.
              These are not tight minimums in all cases; see comments inside code.
              For good performance, LWORK should generally be larger;
              a query is recommended.

    RWORK

              RWORK is REAL array, dimension (MAX(1,LRWORK))
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N',    LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
              else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
              else              LRWORK >= max( 5*mn*mn + 5*mn,
                                               2*mx*mn + 2*mn*mn + mn ).

    IWORK

              IWORK is INTEGER array, dimension (8*min(M,N))

    INFO

              INFO is INTEGER
              <  0:  if INFO = -i, the i-th argument had an illegal value.
              = -4:  if A had a NAN entry.
              >  0:  The updating process of SBDSDC did not converge.
              =  0:  successful exit.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 219 of file cgesdd.f.

  subroutine dgesdd (character jobz, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DGESDD

Purpose:

     DGESDD computes the singular value decomposition (SVD) of a real
     M-by-N matrix A, optionally computing the left and right singular
     vectors.  If singular vectors are desired, it uses a
     divide-and-conquer algorithm.
     The SVD is written
          A = U * SIGMA * transpose(V)
     where SIGMA is an M-by-N matrix which is zero except for its
     min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
     V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
     are the singular values of A; they are real and non-negative, and
     are returned in descending order.  The first min(m,n) columns of
     U and V are the left and right singular vectors of A.
     Note that the routine returns VT = V**T, not V.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'A':  all M columns of U and all N rows of V**T are
                      returned in the arrays U and VT;
              = 'S':  the first min(M,N) columns of U and the first
                      min(M,N) rows of V**T are returned in the arrays U
                      and VT;
              = 'O':  If M >= N, the first N columns of U are overwritten
                      on the array A and all rows of V**T are returned in
                      the array VT;
                      otherwise, all columns of U are returned in the
                      array U and the first M rows of V**T are overwritten
                      in the array A;
              = 'N':  no columns of U or rows of V**T are computed.

    M

              M is INTEGER
              The number of rows of the input matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the input matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if JOBZ = 'O',  A is overwritten with the first N columns
                              of U (the left singular vectors, stored
                              columnwise) if M >= N;
                              A is overwritten with the first M rows
                              of V**T (the right singular vectors, stored
                              rowwise) otherwise.
              if JOBZ .ne. 'O', the contents of A are destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    S

              S is DOUBLE PRECISION array, dimension (min(M,N))
              The singular values of A, sorted so that S(i) >= S(i+1).

    U

              U is DOUBLE PRECISION array, dimension (LDU,UCOL)
              UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
              UCOL = min(M,N) if JOBZ = 'S'.
              If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
              orthogonal matrix U;
              if JOBZ = 'S', U contains the first min(M,N) columns of U
              (the left singular vectors, stored columnwise);
              if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1; if
              JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

    VT

              VT is DOUBLE PRECISION array, dimension (LDVT,N)
              If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
              N-by-N orthogonal matrix V**T;
              if JOBZ = 'S', VT contains the first min(M,N) rows of
              V**T (the right singular vectors, stored rowwise);
              if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1;
              if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
              if JOBZ = 'S', LDVT >= min(M,N).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= 1.
              If LWORK = -1, a workspace query is assumed.  The optimal
              size for the WORK array is calculated and stored in WORK(1),
              and no other work except argument checking is performed.
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
              If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
              If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
              If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
              These are not tight minimums in all cases; see comments inside code.
              For good performance, LWORK should generally be larger;
              a query is recommended.

    IWORK

              IWORK is INTEGER array, dimension (8*min(M,N))

    INFO

              INFO is INTEGER
              <  0:  if INFO = -i, the i-th argument had an illegal value.
              = -4:  if A had a NAN entry.
              >  0:  DBDSDC did not converge, updating process failed.
              =  0:  successful exit.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 211 of file dgesdd.f.

  subroutine sgesdd (character jobz, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

SGESDD

Purpose:

     SGESDD computes the singular value decomposition (SVD) of a real
     M-by-N matrix A, optionally computing the left and right singular
     vectors.  If singular vectors are desired, it uses a
     divide-and-conquer algorithm.
     The SVD is written
          A = U * SIGMA * transpose(V)
     where SIGMA is an M-by-N matrix which is zero except for its
     min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
     V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
     are the singular values of A; they are real and non-negative, and
     are returned in descending order.  The first min(m,n) columns of
     U and V are the left and right singular vectors of A.
     Note that the routine returns VT = V**T, not V.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'A':  all M columns of U and all N rows of V**T are
                      returned in the arrays U and VT;
              = 'S':  the first min(M,N) columns of U and the first
                      min(M,N) rows of V**T are returned in the arrays U
                      and VT;
              = 'O':  If M >= N, the first N columns of U are overwritten
                      on the array A and all rows of V**T are returned in
                      the array VT;
                      otherwise, all columns of U are returned in the
                      array U and the first M rows of V**T are overwritten
                      in the array A;
              = 'N':  no columns of U or rows of V**T are computed.

    M

              M is INTEGER
              The number of rows of the input matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the input matrix A.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if JOBZ = 'O',  A is overwritten with the first N columns
                              of U (the left singular vectors, stored
                              columnwise) if M >= N;
                              A is overwritten with the first M rows
                              of V**T (the right singular vectors, stored
                              rowwise) otherwise.
              if JOBZ .ne. 'O', the contents of A are destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    S

              S is REAL array, dimension (min(M,N))
              The singular values of A, sorted so that S(i) >= S(i+1).

    U

              U is REAL array, dimension (LDU,UCOL)
              UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
              UCOL = min(M,N) if JOBZ = 'S'.
              If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
              orthogonal matrix U;
              if JOBZ = 'S', U contains the first min(M,N) columns of U
              (the left singular vectors, stored columnwise);
              if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1; if
              JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

    VT

              VT is REAL array, dimension (LDVT,N)
              If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
              N-by-N orthogonal matrix V**T;
              if JOBZ = 'S', VT contains the first min(M,N) rows of
              V**T (the right singular vectors, stored rowwise);
              if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1;
              if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
              if JOBZ = 'S', LDVT >= min(M,N).

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= 1.
              If LWORK = -1, a workspace query is assumed.  The optimal
              size for the WORK array is calculated and stored in WORK(1),
              and no other work except argument checking is performed.
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
              If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
              If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
              If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
              These are not tight minimums in all cases; see comments inside code.
              For good performance, LWORK should generally be larger;
              a query is recommended.

    IWORK

              IWORK is INTEGER array, dimension (8*min(M,N))

    INFO

              INFO is INTEGER
              <  0:  if INFO = -i, the i-th argument had an illegal value.
              = -4:  if A had a NAN entry.
              >  0:  SBDSDC did not converge, updating process failed.
              =  0:  successful exit.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 211 of file sgesdd.f.

  subroutine zgesdd (character jobz, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldvt, * ) vt, integer ldvt, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGESDD

Purpose:

     ZGESDD computes the singular value decomposition (SVD) of a complex
     M-by-N matrix A, optionally computing the left and/or right singular
     vectors, by using divide-and-conquer method. The SVD is written
          A = U * SIGMA * conjugate-transpose(V)
     where SIGMA is an M-by-N matrix which is zero except for its
     min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
     V is an N-by-N unitary matrix.  The diagonal elements of SIGMA
     are the singular values of A; they are real and non-negative, and
     are returned in descending order.  The first min(m,n) columns of
     U and V are the left and right singular vectors of A.
     Note that the routine returns VT = V**H, not V.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'A':  all M columns of U and all N rows of V**H are
                      returned in the arrays U and VT;
              = 'S':  the first min(M,N) columns of U and the first
                      min(M,N) rows of V**H are returned in the arrays U
                      and VT;
              = 'O':  If M >= N, the first N columns of U are overwritten
                      in the array A and all rows of V**H are returned in
                      the array VT;
                      otherwise, all columns of U are returned in the
                      array U and the first M rows of V**H are overwritten
                      in the array A;
              = 'N':  no columns of U or rows of V**H are computed.

    M

              M is INTEGER
              The number of rows of the input matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the input matrix A.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if JOBZ = 'O',  A is overwritten with the first N columns
                              of U (the left singular vectors, stored
                              columnwise) if M >= N;
                              A is overwritten with the first M rows
                              of V**H (the right singular vectors, stored
                              rowwise) otherwise.
              if JOBZ .ne. 'O', the contents of A are destroyed.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    S

              S is DOUBLE PRECISION array, dimension (min(M,N))
              The singular values of A, sorted so that S(i) >= S(i+1).

    U

              U is COMPLEX*16 array, dimension (LDU,UCOL)
              UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
              UCOL = min(M,N) if JOBZ = 'S'.
              If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
              unitary matrix U;
              if JOBZ = 'S', U contains the first min(M,N) columns of U
              (the left singular vectors, stored columnwise);
              if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1;
              if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

    VT

              VT is COMPLEX*16 array, dimension (LDVT,N)
              If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
              N-by-N unitary matrix V**H;
              if JOBZ = 'S', VT contains the first min(M,N) rows of
              V**H (the right singular vectors, stored rowwise);
              if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1;
              if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
              if JOBZ = 'S', LDVT >= min(M,N).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= 1.
              If LWORK = -1, a workspace query is assumed.  The optimal
              size for the WORK array is calculated and stored in WORK(1),
              and no other work except argument checking is performed.
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N', LWORK >= 2*mn + mx.
              If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
              If JOBZ = 'S', LWORK >=   mn*mn + 3*mn.
              If JOBZ = 'A', LWORK >=   mn*mn + 2*mn + mx.
              These are not tight minimums in all cases; see comments inside code.
              For good performance, LWORK should generally be larger;
              a query is recommended.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N',    LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
              else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
              else              LRWORK >= max( 5*mn*mn + 5*mn,
                                               2*mx*mn + 2*mn*mn + mn ).

    IWORK

              IWORK is INTEGER array, dimension (8*min(M,N))

    INFO

              INFO is INTEGER
              <  0:  if INFO = -i, the i-th argument had an illegal value.
              = -4:  if A had a NAN entry.
              >  0:  The updating process of DBDSDC did not converge.
              =  0:  successful exit.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 219 of file zgesdd.f.

Author

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