SRC/zgesdd.f(3)

💖 SRC/zgesdd.f 💖

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🌸 NAME 🌸

SRC/zgesdd.f — that’s it, that’s the name. This page lives in the Library Functions Manual. 💅

🎀 SYNOPSIS 🎀

✨ Functions/Subroutines ✨

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

ZGESDD

🦋 Function/Subroutine Documentation 🦋

🌈 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. 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 VH are returned in the array VT; otherwise, all columns of U are returned in the array U and the first M rows of VH 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 VH; if JOBZ = 'S', VT contains the first min(M,N) rows of VH (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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