SRC/zgesvd.f(3)

okay bestie, welcome to the SRC/zgesvd.f(3) page from the LAPACK Library Functions Manual 💅 this one is iconic: it’s the ZGESVD routine, and it’s about to compute the singular value decomposition (SVD) for GE matrices. no cap, it’s giving linear algebra main character energy ✨

NAME 💖

SRC/zgesvd.f — that’s the file, bestie 🎀

SYNOPSIS ✨

Functions/Subroutines 🦋

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

so yeah, ZGESVD computes the singular value decomposition (SVD) for GE matrices 💅

Function/Subroutine Documentation 🦋

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 🔥

Purpose 🌸

okay so here’s the tea: 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 like this:

A = U * SIGMA * conjugate-transpose(V)

where SIGMA is an M-by-N matrix that 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 they are returned in descending order. the first min(m,n) columns of U and V are the left and right singular vectors of A. 🎀

👑 important note bestie: the routine returns V**H, not V.

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.

🚨 warning bestie: JOBVT and JOBU cannot both be 'O'. periodt.

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 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

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) — (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

LDU is INTEGER — the leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', LDU >= M.

VT

VT is 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

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*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 >= 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

RWORK is 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

INFO is 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 💜

Generated automatically by Doxygen for LAPACK from the source code.

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and that’s the whole iconic page, bestie 💖 cute sign-off time: the footer’s source/version is LAPACK Version 3.12.0 with SRC/zgesvd.f(3), and there’s no date printed, so I’m not making one up 🙈🌸 bye bestie, go decompose something responsibly 💅