gesvd(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🦋 gesvd(3) — Library Functions Manual 💅

okay so bestie, welcome to gesvd from the Library Functions Manual — this is the SVD + QR iteration page and it's a long one, so buckle up 🎀

✨ NAME ✨

gesvd — gesvd: SVD, QR iteration. that's it, that's the whole vibe. it does singular value decompositions and it does them correctly.

🌸 SYNOPSIS 🌸

💜 Functions

  • subroutine cgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info) — CGESVD computes the singular value decomposition (SVD) for GE matrices
  • subroutine dgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, info) — DGESVD computes the singular value decomposition (SVD) for GE matrices
  • subroutine sgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, info) — SGESVD computes the singular value decomposition (SVD) for GE matrices
  • subroutine zgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info) — ZGESVD computes the singular value decomposition (SVD) for GE matrices

🌈 Detailed Description 🌈

no extra prose here bestie, it goes straight into the good stuff 👇

👑 Function Documentation 👑

🌟 subroutine cgesvd

subroutine cgesvd (character jobu, character jobvt, 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 info)

CGESVD computes the singular value decomposition (SVD) for GE matrices — periodt.

🧚 Purpose:

okay so this is the moment. CGESVD computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written like this, iconic math incoming:

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 V**H, not V. not me warning you about this, it's literally the whole personality of this function 💀

🌸 Parameters

  • JOBU — JOBU is CHARACTER*1. Specifies options for computing all or part of the matrix U:
    • = 'A': all M columns of U are returned in array U:
    • = 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U;
    • = 'O': the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A;
    • = 'N': no columns of U (no left singular vectors) are computed.
  • JOBVT — JOBVT is CHARACTER*1. Specifies options for computing all or part of the matrix V**H:
    • = 'A': all N rows of V**H are returned in the array VT;
    • = 'S': the first min(m,n) rows of V**H (the right singular vectors) are returned in the array VT;
    • = 'O': the first min(m,n) rows of V**H (the right singular vectors) are overwritten on the array A;
    • = 'N': no rows of V**H (no right singular vectors) are computed.
⚠️ JOBVT and JOBU cannot both be 'O'. like, literally cannot, don't even try it 🙈
  • 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 JOBU = 'O', A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBVT = 'O', A is overwritten with the first min(m,n) rows of V**H (the right singular vectors, stored rowwise);
    • if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A are destroyed.
💀 hi yes, "destroyed" is the technical term. your beautiful matrix is gone. she served her purpose.
  • 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). descending order, the way nature intended ✨
  • U — U is COMPLEX array, dimension (LDU,UCOL) — that's (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU = 'A', U contains the M-by-M unitary matrix U; if JOBU = 'S', U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise); if JOBU = 'N' or 'O', U is not referenced. she's simply not part of the conversation 😌
  • LDU — LDU is INTEGER. The leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', LDU >= M.
  • VT — VT is COMPLEX array, dimension (LDVT,N). If JOBVT = 'A', VT contains the N-by-N unitary matrix V**H; if JOBVT = 'S', VT contains the first min(m,n) rows of V**H (the right singular vectors, stored rowwise); if JOBVT = 'N' or 'O', VT is not referenced.
  • LDVT — LDVT is INTEGER. The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'A', LDVT >= N; if JOBVT = '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. free real estate 🫶
  • LWORK — LWORK is INTEGER. The dimension of the array WORK.
    • LWORK >= MAX(1,2*MIN(M,N)+MAX(M,N)).
    • For good performance, LWORK should generally be larger. (bestie. give her room to breathe. she deserves it.)
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • RWORK — RWORK is REAL array, dimension (5*min(M,N)). On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A = U * B * VT, so it has the same singular values as A, and singular vectors related by U and VT. iykyk 🔥
  • INFO — INFO is INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: if CBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero. See the description of RWORK above for details.

🎀 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 212 of file cgesvd.f.

💫 subroutine dgesvd

subroutine dgesvd (character jobu, character jobvt, 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 info)

DGESVD computes the singular value decomposition (SVD) for GE matrices — same energy, but make it real (literally, it's the double precision one) 💖

🧚 Purpose:

DGESVD computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. 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 V**T, not V. (the complex sibling returns V**H — same reason, different conjugation situation. the way that these two are cousins 😭)

🌸 Parameters

  • JOBU — JOBU is CHARACTER*1. Specifies options for computing all or part of the matrix U:
    • = 'A': all M columns of U are returned in array U:
    • = 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U;
    • = 'O': the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A;
    • = 'N': no columns of U (no left singular vectors) are computed.
  • JOBVT — JOBVT is CHARACTER*1. Specifies options for computing all or part of the matrix V**T:
    • = 'A': all N rows of V**T are returned in the array VT;
    • = 'S': the first min(m,n) rows of V**T (the right singular vectors) are returned in the array VT;
    • = 'O': the first min(m,n) rows of V**T (the right singular vectors) are overwritten on the array A;
    • = 'N': no rows of V**T (no right singular vectors) are computed.
⚠️ JOBVT and JOBU cannot both be 'O'. same rule, still not negotiable 🚫
  • 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 — okay so A is a DOUBLE PRECISION array with dimension (LDA,N). On entry, it’s the M-by-N matrix A. On exit? It depends, bestie:
    • if JOBU = 'O', A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBVT = 'O', A is overwritten with the first min(m,n) rows of V**T (the right singular vectors, stored rowwise);
    • if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A are destroyed. Yep, gone. 💀
  • LDA — LDA is INTEGER, the leading dimension of the array A. Constraint: LDA >= max(1,M). Periodt.
  • S — S is DOUBLE PRECISION array, dimension (min(M,N)). It holds the singular values of A, sorted so that S(i) >= S(i+1). Slay, descending order. ✨
  • U — U is DOUBLE PRECISION array, dimension (LDU,UCOL) — that’s (LDU,M) if JOBU = 'A', or (LDU,min(M,N)) if JOBU = 'S'. Specifically:
    • if JOBU = 'A', U contains the M-by-M orthogonal matrix U;
    • if JOBU = 'S', U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBU = 'N' or 'O', U is not referenced. Not even looked at. 🙈
  • LDU — LDU is INTEGER, the leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', then LDU >= M.
  • VT — VT is DOUBLE PRECISION array, dimension (LDVT,N).
    • If JOBVT = 'A', VT contains the N-by-N orthogonal matrix V**T;
    • if JOBVT = 'S', VT contains the first min(m,n) rows of V**T (the right singular vectors, stored rowwise);
    • if JOBVT = 'N' or 'O', VT is not referenced.
  • LDVT — LDVT is INTEGER, the leading dimension of the array VT. LDVT >= 1; if JOBVT = 'A', LDVT >= N; if JOBVT = '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; if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). And B satisfies A = U * B * VT, so it has the same singular values as A, and singular vectors related by U and VT. Iconic math behavior. 🧚
  • LWORK — LWORK is INTEGER, the dimension of the array WORK. Here are the requirements:
    • LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
      • PATH 1 (M much larger than N, JOBU='N')
      • PATH 1t (N much larger than M, JOBVT='N')
    • LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(M,N)) for the other paths.
    • For good performance, LWORK should generally be larger. Like, bigger is better, no cap.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. So it’s just reporting the vibe, not doing the whole job. 🫶
  • INFO — INFO is INTEGER:
    • = 0: successful exit. 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. Rude.
    • > 0: if DBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero. See the description of WORK above for details.

👑 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 209 of file dgesvd.f.

💫 subroutine sgesvd 💫

   subroutine sgesvd (character jobu, character jobvt, 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 info)

SGESVD computes the singular value decomposition (SVD) for GE matrices. Main character energy. 👑

🎯 Purpose

Okay so SGESVD computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. 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 V**T, not V. Do not come for V; it’s giving transpose. 💅

🌸 Parameters

  • JOBU — JOBU is CHARACTER*1. It specifies options for computing all or part of the matrix U:
    • = 'A': all M columns of U are returned in array U;
    • = 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U;
    • = 'O': the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A;
    • = 'N': no columns of U (no left singular vectors) are computed.
  • JOBVT — JOBVT is CHARACTER*1. It specifies options for computing all or part of the matrix V**T:
    • = 'A': all N rows of V**T are returned in the array VT;
    • = 'S': the first min(m,n) rows of V**T (the right singular vectors) are returned in the array VT;
    • = 'O': the first min(m,n) rows of V**T (the right singular vectors) are overwritten on the array A;
    • = 'N': no rows of V**T (no right singular vectors) are computed.
JOBVT and JOBU cannot both be 'O'. Like, pick a lane, bestie. 🚫
  • 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 JOBU = 'O', A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBVT = 'O', A is overwritten with the first min(m,n) rows of V**T (the right singular vectors, stored rowwise);
    • if JOBU .ne. 'O' and JOBVT .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)). These are the singular values of A, sorted so that S(i) >= S(i+1).
  • U — U is REAL array, dimension (LDU,UCOL) — (LDU,M) if JOBU = 'A', or (LDU,min(M,N)) if JOBU = 'S'.
    • If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
    • if JOBU = 'S', U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBU = 'N' or 'O', U is not referenced.
  • LDU — LDU is INTEGER, the leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', LDU >= M.
  • VT — VT is REAL array, dimension (LDVT,N).
    • If JOBVT = 'A', VT contains the N-by-N orthogonal matrix V**T;
    • if JOBVT = 'S', VT contains the first min(m,n) rows of V**T (the right singular vectors, stored rowwise);
    • if JOBVT = 'N' or 'O', VT is not referenced.
  • LDVT — LDVT is INTEGER, the leading dimension of the array VT. LDVT >= 1; if JOBVT = 'A', LDVT >= N; if JOBVT = '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; if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A = U * B * VT, so it has the same singular values as A, and singular vectors related by U and VT. Yep, REAL this time. 🍓
  • LWORK — LWORK is INTEGER, the dimension of the array WORK. Requirements:
    • LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
      • PATH 1 (M much larger than N, JOBU='N')
      • PATH 1t (N much larger than M, JOBVT='N')
    • LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) for the other paths.
    • For good performance, LWORK should generally be larger.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INFO is INTEGER:
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: if SBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero. See the description of WORK above for details.

👑 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 209 of file sgesvd.f.

subroutine zgesvd (character jobu, character jobvt, 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 info)

ZGESVD computes the singular value decomposition (SVD) for GE matrices. Bestie, this is the one that decomposes your complex matrix and can hand you back the singular vectors if you ask nicely 💅

Purpose

okay so ZGESVD computes the singular value decomposition (SVD) of a complex M-by-N matrix A, and optionally computes the left and/or right singular vectors. 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 V**H, not V. that's the plot twist, bestie.

Parameters

  • JOBU — CHARACTER*1. This tells the routine what to do about all or part of the matrix U:
    • 'A': all M columns of U are returned in array U:
    • 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U;
    • 'O': the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A;
    • 'N': no columns of U (no left singular vectors) are computed.
  • JOBVT — CHARACTER*1. This tells the routine what to do about all or part of the matrix V**H:
    • 'A': all N rows of V**H are returned in the array VT;
    • 'S': the first min(m,n) rows of V**H (the right singular vectors) are returned in the array VT;
    • 'O': the first min(m,n) rows of V**H (the right singular vectors) are overwritten on the array A;
    • 'N': no rows of V**H (no right singular vectors) are computed.
🚨 bestie, JOBVT and JOBU cannot both be 'O'. that combo is not it.
  • M — INTEGER. The number of rows of the input matrix A. M >= 0.
  • N — INTEGER. The number of columns of the input matrix A. N >= 0.
  • A — COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit:
    • if JOBU = 'O', A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBVT = 'O', A is overwritten with the first min(m,n) rows of V**H (the right singular vectors, stored rowwise);
    • if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A are destroyed.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • S — DOUBLE PRECISION array, dimension (min(M,N)). The singular values of A, sorted so that S(i) >= S(i+1).
  • U — COMPLEX*16 array, dimension (LDU,UCOL) — (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
    • If JOBU = 'A', U contains the M-by-M unitary matrix U;
    • if JOBU = 'S', U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise);
    • if JOBU = 'N' or 'O', U is not referenced.
  • LDU — INTEGER. The leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', LDU >= M.
  • VT — COMPLEX*16 array, dimension (LDVT,N).
    • If JOBVT = 'A', VT contains the N-by-N unitary matrix V**H;
    • if JOBVT = 'S', VT contains the first min(m,n) rows of V**H (the right singular vectors, stored rowwise);
    • if JOBVT = 'N' or 'O', VT is not referenced.
  • LDVT — INTEGER. The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
  • WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — INTEGER. The dimension of the array WORK.
    • LWORK >= MAX(1,2*MIN(M,N)+MAX(M,N)).
    • For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • RWORK — DOUBLE PRECISION array, dimension (5*min(M,N)). On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A = U * B * VT, so it has the same singular values as A, and singular vectors related by U and VT.
  • INFO — INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: if ZBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero. See the description of RWORK above for details.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 212 of file zgesvd.f.

Author

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