lasd1(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💅 NAME 💅

okay so bestie, this is lasd1(3) and the whole vibe is: lasd1 — lasd1: D&C step: merge subproblems. that's the main character energy here, no cap ✨

✨ SYNOPSIS ✨

🧚 Functions 🧚

dlasd1 — DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.</p>

subroutine dlasd1 (nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info)

slasd1 — SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.</p>

subroutine slasd1 (nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info)

🦋 Detailed Description 🦋

the real tea is in the function docs right below, bestie. iykyk 💅

🎀 Function Documentation 🎀

💖 subroutine dlasd1 💖

subroutine dlasd1 (integer nl, integer nr, integer sqre, double precision, dimension( * ) d, double precision alpha, double precision beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * ) idxq, integer, dimension( * ) iwork, double precision, dimension( * ) work, integer info)

🌸 Purpose 🌸

okay so DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.

a related subroutine DLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired.

DLASD1 computes the SVD as follows:

              ( D1(in)    0    0       0 )
  B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
              (   0       0   D2(in)   0 )

    = U(out) * ( D(out) 0) * VT(out)

where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0.

the left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. the algorithm consists of three stages:

the first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. for each such occurrence the dimension of the secular equation problem is reduced by one. this stage is performed by the routine DLASD2.
the second stage consists of calculating the updated singular values. this is done by finding the square roots of the roots of the secular equation via the routine DLASD4 (as called by DLASD3). this routine also calculates the singular vectors of the current problem.
the final stage consists of computing the updated singular vectors directly using the updated singular values. the singular vectors for the current problem are multiplied with the singular vectors from the overall problem.

🌸 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 row dimension N = NL + NR + 1, and column dimension M = N + SQRE.
D — DOUBLE PRECISION array, dimension (N = NL+NR+1). on entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. on exit D(1:N) contains the singular values of the modified matrix.
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(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. on exit U contains the left singular vectors of the bidiagonal matrix.
LDU — INTEGER. the leading dimension of the array U. LDU >= max( 1, N ).
VT — DOUBLE PRECISION array, dimension(LDVT,M) where M = N + SQRE. on entry VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. on exit VT**T contains the right singular vectors of the bidiagonal matrix.
LDVT — INTEGER. the leading dimension of the array VT. LDVT >= max( 1, M ).
IDXQ — INTEGER array, dimension(N). this contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order.
IWORK — INTEGER array, dimension( 4 * N ).
WORK — DOUBLE PRECISION array, dimension( 3*M**2 + 2*M ).
INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, a singular value did not converge.

👑 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 202 of file dlasd1.f.

🦋 subroutine slasd1 🦋

subroutine slasd1 (integer nl, integer nr, integer sqre, real, dimension( * ) d, real alpha, real beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * ) idxq, integer, dimension( * ) iwork, real, dimension( * ) work, integer info)

🌸 Purpose 🌸

okay so SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.

a related subroutine SLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired.

SLASD1 computes the SVD as follows:

              ( D1(in)    0    0       0 )
  B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
              (   0       0   D2(in)   0 )

    = U(out) * ( D(out) 0) * VT(out)

the first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. for each such occurrence the dimension of the secular equation problem is reduced by one. this stage is performed by the routine SLASD2.
the second stage consists of calculating the updated singular values. this is done by finding the square roots of the roots of the secular equation via the routine SLASD4 (as called by SLASD3). this routine also calculates the singular vectors of the current problem.
the final stage consists of computing the updated singular vectors directly using the updated singular values. the singular vectors for the current problem are multiplied with the singular vectors from the overall problem.

🌸 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 row dimension N = NL + NR + 1, and column dimension M = N + SQRE.
D — REAL array, dimension (NL+NR+1). N = NL+NR+1. on entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. on exit D(1:N) contains the singular values of the modified matrix.
ALPHA — REAL. contains the diagonal element associated with the added row.
BETA — REAL. contains the off-diagonal element associated with the added row.
U — REAL array, dimension (LDU,N). on entry U(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. on exit U contains the left singular vectors of the bidiagonal matrix.
LDU — INTEGER. the leading dimension of the array U. LDU >= max( 1, N ).
VT

okay so we're still in the argument tea, bestie — last little stack, every deet stays, no cap:

VT — VT is REAL array, dimension (LDVT,M), where M = N + SQRE. On entry, VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. On exit, VT**T contains the right singular vectors of the bidiagonal matrix. literally the container for the singular-vector glow-up.
LDVT — LDVT is INTEGER. It's the leading dimension of the array VT. Constraint: LDVT >= max( 1, M ). respect the leading dimension, bestie.
IDXQ — IDXQ is INTEGER array, dimension (N). It contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order. sorting tea, served fresh.
IWORK — IWORK is INTEGER array, dimension (4*N). integer workspace, periodt.
WORK — WORK is REAL array, dimension (3*M**2+2*M). real workspace, iconic.
INFO — INFO is INTEGER:
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = 1, a singular value did not converge.

💖 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 202 of file slasd1.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

byeee bestie — footer tea is LAPACK, Version 3.12.0, page lasd1(3) (no date line in this one). mwah 💋✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms