lasd2(3)

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💅 lasd2(3) — lasd2: D&C step: deflation ✨

(Library Functions Manual, but make it group chat.)

🌸 NAME 🌸

lasd2 — lasd2: D&C step: deflation

🫶 SYNOPSIS 🫶

Functions 💖

dlasd2 🦋

subroutine dlasd2 (nl, nr, sqre, k, d, z, alpha, beta, u, ldu, vt, ldvt,
    dsigma, u2, ldu2, vt2, ldvt2, idxp, idx, idxc, idxq, coltyp, info)

DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.

slasd2 🌸

subroutine slasd2 (nl, nr, sqre, k, d, z, alpha, beta, u, ldu, vt, ldvt,
    dsigma, u2, ldu2, vt2, ldvt2, idxp, idx, idxc, idxq, coltyp, info)

SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.

🦋 Detailed Description 🦋

Okay bestie, this page is the function-docs era of lasd2 — two subroutines, one very detailed vibe. Let’s get into it 💅

Function Documentation 💖

dlasd2 — the DOUBLE PRECISION one ✨

subroutine dlasd2 (integer nl, integer nr, integer sqre, integer k, double
    precision, dimension( * ) d, double precision, dimension( * ) z, double
    precision alpha, double precision beta, double precision, dimension(
    ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt,
    integer ldvt, double precision, dimension( * ) dsigma, double precision,
    dimension( ldu2, * ) u2, integer ldu2, double precision, dimension(
    ldvt2, * ) vt2, integer ldvt2, integer, dimension( * ) idxp, integer,
    dimension( * ) idx, integer, dimension( * ) idxc, integer, dimension( *
    ) idxq, integer, dimension( * ) coltyp, integer info)

DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.

Purpose 🌟

Bestie, here’s the whole point: DLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

DLASD2 is called from DLASD1.

Parameters 🎀

  • NL — INTEGER. The row dimension of the upper block. NL >= 1.
  • NR — INTEGER. The row dimension of the lower block. NR >= 1.
  • SQRE — INTEGER. = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.
  • K — INTEGER. Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.
  • D — DOUBLE PRECISION array, dimension (N). On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order.
  • Z — DOUBLE PRECISION array, dimension (N). On exit Z contains the updating row vector in the secular equation.
  • ALPHA — DOUBLE PRECISION. Contains the diagonal element associated with the added row.
  • BETA — DOUBLE PRECISION. Contains the off-diagonal element associated with the added row.
  • U — DOUBLE PRECISION array, dimension (LDU,N). On entry U contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (N-K) updated left singular vectors (those which were deflated) in its last N-K columns.
  • LDU — INTEGER. The leading dimension of the array U. LDU >= N.
  • VT — DOUBLE PRECISION array, dimension (LDVT,M). On entry VT**T contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT**T contains the trailing (N-K) updated right singular vectors (those which were deflated) in its last N-K columns. In case SQRE =1, the last row of VT spans the right null space.
  • LDVT — INTEGER. The leading dimension of the array VT. LDVT >= M.
  • DSIGMA — DOUBLE PRECISION array, dimension (N). Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation.
  • U2 — DOUBLE PRECISION array, dimension (LDU2,N). Contains a copy of the first K-1 left singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains non-zero entries only at and above NL; the third contains non-zero entries only below NL+1; and the fourth is dense.
  • LDU2 — INTEGER. The leading dimension of the array U2. LDU2 >= N.
  • VT2 — DOUBLE PRECISION array, dimension (LDVT2,N). VT2**T contains a copy of the first K right singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains non-zeros only at and before NL +1; the third block contains non-zeros only at and after NL +2.
  • LDVT2 — INTEGER. The leading dimension of the array VT2. LDVT2 >= M.
  • IDXP — INTEGER array, dimension (N). This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values.
  • IDX — INTEGER array, dimension (N). This will contain the permutation used to sort the contents of D into ascending order.
  • IDXC — INTEGER array, dimension (N). This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains non-zero entries only at and above NL, the second contains non-zero entries only below NL+2, and the third is dense.
  • IDXQ — INTEGER array, dimension (N). This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values.
  • COLTYP — INTEGER array, dimension (N). As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is:
    1. 1 : non-zero in the upper half only
    2. 2 : non-zero in the lower half only
    3. 3 : dense
    4. 4 : deflated

On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the I-th type columns.

  • INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

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 266 of file dlasd2.f.

slasd2 — the real one 🌈

subroutine slasd2 (integer nl, integer nr, integer sqre, integer k, real,
    dimension( * ) d, real, dimension( * ) z, real alpha, real beta, real,
    dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt,
    integer ldvt, real, dimension( * ) dsigma, real, dimension( ldu2, * )
    u2, integer ldu2, real, dimension( ldvt2, * ) vt2, integer ldvt2,
    integer, dimension( * ) idxp, integer, dimension( * ) idx, integer,
    dimension( * ) idxc, integer, dimension( * ) idxq, integer, dimension( *
    ) coltyp, integer info)

SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.

Purpose 🌟

SLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.

SLASD2 is called from SLASD1.

Parameters 🎀

  • NL — INTEGER. The row dimension of the upper block. NL >= 1.
  • NR — INTEGER. The row dimension of the lower block. NR >= 1.
  • SQRE — INTEGER. = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.
  • K — INTEGER. Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.

parameter parade, part two — every detail stays, only the vibe changes 💅

  • D — D is REAL array, dimension (N). On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order. ✨
  • Z — Z is REAL array, dimension (N). On exit Z contains the updating row vector in the secular equation. 🫶
  • ALPHA — ALPHA is REAL. Contains the diagonal element associated with the added row. 💖
  • BETA — BETA is REAL. Contains the off-diagonal element associated with the added row. 🎀
  • U — U is REAL array, dimension (LDU,N). On entry U contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (N-K) updated left singular vectors (those which were deflated) in its last N-K columns. 🦋
  • LDU — LDU is INTEGER. The leading dimension of the array U. LDU >= N. 👑
  • VT — VT is REAL array, dimension (LDVT,M). On entry VT**T contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT**T contains the trailing (N-K) updated right singular vectors (those which were deflated) in its last N-K columns. In case SQRE =1, the last row of VT spans the right null space. 🌸
  • LDVT — LDVT is INTEGER. The leading dimension of the array VT. LDVT >= M. 💅
  • DSIGMA — DSIGMA is REAL array, dimension (N). Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation. 🍓
  • U2 — U2 is REAL array, dimension (LDU2,N). Contains a copy of the first K-1 left singular vectors which will be used by SLASD3 in a matrix multiply (SGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains non-zero entries only at and above NL; the third contains non-zero entries only below NL+1; and the fourth is dense. 🔥
  • LDU2 — LDU2 is INTEGER. The leading dimension of the array U2. LDU2 >= N. 🌈
  • VT2 — VT2 is REAL array, dimension (LDVT2,N). VT2**T contains a copy of the first K right singular vectors which will be used by SLASD3 in a matrix multiply (SGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains non-zeros only at and before NL +1; the third block contains non-zeros only at and after NL +2. ⭐
  • LDVT2 — LDVT2 is INTEGER. The leading dimension of the array VT2. LDVT2 >= M. 🧚
  • IDXP — IDXP is INTEGER array, dimension (N). This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values. 💜
  • IDX — IDX is INTEGER array, dimension (N). This will contain the permutation used to sort the contents of D into ascending order. 🙈
  • IDXC — IDXC is INTEGER array, dimension (N). This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains non-zero entries only at and above NL, the second contains non-zero entries only below NL+2, and the third is dense. 💀
  • IDXQ — IDXQ is INTEGER array, dimension (N). This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values. (Yes, hlaf is the source spelling — not me, bestie 😭)
  • COLTYP — COLTYP is INTEGER array, dimension (N). As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is: 1 : non-zero in the upper half only, 2 : non-zero in the lower half only, 3 : dense, 4 : deflated. On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the I-th type columns. 🎀
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. 💖

👑 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 266 of file slasd2.f. ✨

Author (Doxygen's encore)

Generated automatically by Doxygen for LAPACK from the source code. 🧚

🎀 footer tea, bestie: source/version is LAPACK Version 3.12.0, page tag is lasd2(3) — the footer doesn't list a date, so that's the whole sign-off, periodt 💖

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