gesdd(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

🌸 gesdd(3) — Library Functions Manual 🌸

Bestie, this is the gesdd(3) page, and its NAME is literally gesdd — gesdd: SVD, divide and conquer. We're doing the whole group-chat walkthrough, no cap. 💅✨

🎀 SYNOPSIS 🎀

🫶 Functions 🫶

CGESDD — the complex version, okay?

       subroutine cgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork,
           rwork, iwork, info)
           CGESDD

DGESDD — the double precision version.

       subroutine dgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork,
           iwork, info)
           DGESDD

SGESDD — the single precision version.

       subroutine sgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork,
           iwork, info)
           SGESDD

ZGESDD — the complex double precision version.

       subroutine zgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork,
           rwork, iwork, info)
           ZGESDD

💫 DETAILED DESCRIPTION 💫

Okay so the detailed description is basically the function docs below, bestie. Let's get into it. 🦋

🎀 FUNCTION DOCUMENTATION 🎀

💅 `CGESDD` — subroutine cgesdd 💅

   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 the 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. Iconic behavior, honestly. 👑

🌈 Parameters 🌈

JOBZ — JOBZ is CHARACTER*1. It 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.

Bestie, clock that warning: if JOBZ .ne. 'O', the contents of A are destroyed. Not a drill. 💀

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.

🔥 `DGESDD` — subroutine dgesdd 🔥

   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. Again: VT is the moment, not V itself. 💅

🌈 Parameters 🌈

JOBZ — JOBZ is CHARACTER*1. It 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.

Okay but seriously, bestie: if JOBZ .ne. 'O', the contents of A are destroyed. Protect your data. 😭

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 — okay so U is a 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. It’s the leading dimension of the array U. LDU >= 1; if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, then 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. Leading dimension of 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. 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

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

okay bestie, 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. 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. 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. 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

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 — the main character option: it specifies options for computing all or part of the matrix U, and it's giving choose-your-own-adventure 💅
- '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 is INTEGER — the number of rows of the input matrix A. M >= 0.
N is INTEGER — the number of columns of the input matrix A. N >= 0.
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. Destroyed, bestie — not preserved.
LDA is INTEGER — the leading dimension of the array A. LDA >= max(1,M).
S is DOUBLE PRECISION array, dimension (min(M,N)) — the singular values of A, sorted so that S(i) >= S(i+1).
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 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 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 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 is COMPLEX*16 array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, WORK(1) returns the optimal 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. Don't skimp on the workspace, bestie — she needs room.
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 is INTEGER array, dimension (8*min(M,N)).
INFO is INTEGER — the exit-status gossip:
- INFO < 0: if INFO = -i, the i-th argument had an illegal value.
- INFO = -4: if A had a NAN entry.
- INFO > 0: The updating process of DBDSDC did not converge.
- INFO = 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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gesdd(3)

🌸 gesdd(3) — Library Functions Manual 🌸

🎀 SYNOPSIS 🎀

🫶 Functions 🫶

💫 DETAILED DESCRIPTION 💫

🎀 FUNCTION DOCUMENTATION 🎀

💅 CGESDD — subroutine cgesdd 💅

✨ Purpose ✨

🌈 Parameters 🌈

👑 Author 👑

🌈 Contributors: 🌈

🔥 DGESDD — subroutine dgesdd 🔥

✨ Purpose ✨

🌈 Parameters 🌈

Author

Contributors

subroutine sgesdd

Purpose

Parameters

Author

Contributors

subroutine zgesdd

Purpose

Parameters

👑 Author

🌟 Contributors

🤖 Author

💅 `CGESDD` — subroutine cgesdd 💅

🔥 `DGESDD` — subroutine dgesdd 🔥