SRC/sgesdd.f(3)

🌸 NAME 🌸

Hey bestie, this is the whole vibe for SRC/sgesdd.f(3) from the LAPACK Library Functions Manual 💅✨. It's the page for SGESDD, and it's honestly main character energy.

✨ SYNOPSIS ✨

Functions/Subroutines

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

SGESDD 🫶

🦋 Function/Subroutine Documentation 🦋

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 so SGESDD computes the singular value decomposition (SVD) of a real M-by-N matrix A, and it can optionally compute the left and right singular vectors. If singular vectors are desired, it uses a divide-and-conquer algorithm. Slay.

The SVD is written:

A = U * SIGMA * transpose(V)

Here 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. Big important vibe check 💖.

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 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. Yay ✨

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.

Author

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That's the whole page, bestie 🫶. It's giving LAPACK Version 3.12.0, source file SRC/sgesdd.f(3), and the final sign-off is pure "generated automatically by Doxygen for LAPACK from the source code" energy ✨👑 periodt.