gelsd(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 gelsd(3) 💖 — least squares using SVD, divide and conquer

From the Library Functions Manual, bestie — this is the gelsd(3) glow-up. We're doing least squares with SVD and divide-and-conquer energy, and yes, it's iconic. ✨

✨ NAME ✨

gelsd - gelsd: least squares using SVD, divide and conquer

🦋 SYNOPSIS 🦋

🧚 Functions 🧚

  • subroutine cgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info) — CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
  • subroutine dgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info) — DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
  • subroutine sgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info) — SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
  • subroutine zgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info) — ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

🌈 DETAILED DESCRIPTION 🌈

This heading is literally all that's here in part 1 — the original page then jumps straight into Function Documentation. No cap. 🦋

👑 FUNCTION DOCUMENTATION 👑

🫶 cgelsd 🫶

subroutine cgelsd (integer m, integer n, integer nrhs, complex, dimension(
    lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,
    real, dimension( * ) s, real rcond, integer rank, complex, dimension( *
    ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( *
    ) iwork, integer info)

CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices.

🎯 Purpose 🎯

CGELSD computes the minimum-norm solution to a real linear least squares problem:

minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. 🫶

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The problem is solved in three steps:

(1) Reduce the coefficient matrix A to bidiagonal form with
    Householder transformations, reducing the original problem
    into a  (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder transformations to solve
    the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

🍓 Parameters 🍓

M — M is INTEGER. It's the number of rows of the matrix A. Constraint: M >= 0.

N — N is INTEGER. It's the number of columns of the matrix A. Constraint: N >= 0.

NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. Constraint: NRHS >= 0.

A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, A has been destroyed. 💀 Yes, gone. That's the vibe.

LDA — LDA is INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,M).

B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of the modulus of elements n+1:m in that column. 🎀

LDB — LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,M,N).

S — S is REAL array, dimension (min(M,N)). The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND — RCOND is REAL. RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.

RANK — RANK is INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).

WORK — WORK is COMPLEX 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 must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least

2 * N + N * NRHS

if M is greater than or equal to N or

2 * M + M * NRHS

if M is less than N, the code will execute correctly. 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 array WORK and the minimum sizes of the arrays RWORK and IWORK, and returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA.

RWORK — RWORK is REAL array, dimension (MAX(1,LRWORK)). The requirement is:

LRWORK >=
   10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
   MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )

if M is greater than or equal to N or

   10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
   MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )

if M is less than N, the code will execute correctly.

SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and

NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). Constraint: LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO — INFO is INTEGER.

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

🌟 Author 🌟

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

💜 Contributors 💜

  • Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
  • Osni Marques, LBNL/NERSC, USA

Definition at line 217 of file cgelsd.f.

💜 dgelsd 💜

subroutine dgelsd (integer m, integer n, integer nrhs, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *
    ) b, integer ldb, double precision, dimension( * ) s, double precision
    rcond, integer rank, double precision, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer info)

DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices.

🎯 Purpose 🎯

DGELSD computes the minimum-norm solution to a real linear least squares problem:

minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. ✨

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The problem is solved in three steps:

(1) Reduce the coefficient matrix A to bidiagonal form with
    Householder transformations, reducing the original problem
    into a  (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder transformations to solve
    the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

🍓 Parameters 🍓

M — M is INTEGER. The number of rows of A. Constraint: M >= 0.

N — N is INTEGER. The number of columns of A. Constraint: N >= 0.

NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. Constraint: NRHS >= 0.

A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, A has been destroyed. 💀

LDA — LDA is INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,M).

B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. 🎀

LDB — LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,max(M,N)).

S — S is DOUBLE PRECISION array, dimension (min(M,N)). The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND — (description continues in part 2, bestie 💅)

  • RCOND — DOUBLE PRECISION, bestie. It’s the cutoff for the effective rank of A: singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead. Slay. ✨
  • RANK — INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1). That’s the whole vibe check right there. 👑
  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Iconic little self-report, honestly. 💅
  • LWORK — INTEGER. The dimension of the array WORK. LWORK must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ). 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. Not me knowing all these workspace formulas by heart 😭💖
  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN), where MINMN = MIN( M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. Cute, it also reports back. 🫶
  • INFO — INTEGER. The exit-status tea:
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  the algorithm for computing the SVD failed to converge;
      if INFO = i, i off-diagonal elements of an intermediate
      bidiagonal form did not converge to zero.

💖 Author 💖

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🫶 Contributors 🫶

  • Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
  • Osni Marques, LBNL/NERSC, USA

Definition at line 201 of file dgelsd.f.

🌸 subroutine sgelsd 🌸

   subroutine sgelsd (integer m, integer n, integer nrhs, real, dimension( lda,
       * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
       dimension( * ) s, real rcond, integer rank, real, dimension( * ) work,
       integer lwork, integer, dimension( * ) iwork, integer info)

SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices.

✨ Purpose ✨

Okay so SGELSD computes the minimum-norm solution to a real linear least squares problem:

minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The problem is solved in three steps:

  1. Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a (BLS) — yes, that’s literally the source text, bestie 🫶
  2. Solve the BLS using a divide and conquer approach.
  3. Apply back all the Householder transformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

🎀 Parameters 🎀

  • M — INTEGER. The number of rows of A. M >= 0. Rows are having a main character moment. 🌈
  • N — INTEGER. The number of columns of A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • A — REAL array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, A has been destroyed. Not “modified,” bestie — destroyed 💀
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — REAL array, dimension (LDB,NRHS). On entry, the M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. Little glow-up moment for B. ✨
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,max(M,N)).
  • S — REAL array, dimension (min(M,N)). The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)). Math girlies rise. 🎀
  • RCOND — REAL. RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.
  • RANK — INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — INTEGER. The dimension of the array WORK. LWORK must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ). 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 array WORK and the minimum size of the array IWORK, and returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA. She’s thorough, okay? 💅
  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
  • INFO — INTEGER. The exit-status tea:
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  the algorithm for computing the SVD failed to converge;
      if INFO = i, i off-diagonal elements of an intermediate
      bidiagonal form did not converge to zero.

💖 Author 💖

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🫶 Contributors 🫶

  • Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
  • Osni Marques, LBNL/NERSC, USA

Definition at line 202 of file sgelsd.f.

💜 subroutine zgelsd 💜

   subroutine zgelsd (integer m, integer n, integer nrhs, complex*16,
       dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
       integer ldb, double precision, dimension( * ) s, double precision rcond,
       integer rank, complex*16, dimension( * ) work, integer lwork, double
       precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer
       info)

ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices.

⭐ Purpose ⭐

ZGELSD computes the minimum-norm solution to a real linear least squares problem:

minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The problem is solved in three steps:

  1. Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a (BLS) — same exact source text, no cap 🫶
  2. Solve the BLS using a divide and conquer approach.
  3. Apply back all the Householder transformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

🦋 Parameters 🦋

  • M — INTEGER. The number of rows of the matrix A. M >= 0.
  • N — INTEGER. The number of columns of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • A —

A

A is COMPLEX*16 array, dimension (LDA,N) 💅 on entry it's the M-by-N matrix A, and on exit — bestie, brace yourself — A has been destroyed. Gone. Reduced to atoms. She served her purpose 😭

LDA

LDA is INTEGER. She's the leading dimension of the array A, and the vibe check is LDA >= max(1,M) 🫶

B

B is COMPLEX*16 array, dimension (LDB,NRHS). On entry she's the M-by-NRHS right hand side matrix B, and on exit she gets overwritten by the N-by-NRHS solution matrix X — glow-up, honestly ✨

Also, tiny detail that actually matters: if m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of the modulus of elements n+1:m in that column 🍓

LDB

LDB is INTEGER. Leading dimension of the array B, and the rule is LDB >= max(1,M,N) — yes, both of them, no cutting corners 💀

S

S is DOUBLE PRECISION array, dimension (min(M,N)). She holds the singular values of A in decreasing order 👑

And the condition number of A in the 2-norm = S(1)/S(min(m,n)) — iconic little formula, iykyk 🌸

RCOND

RCOND is DOUBLE PRECISION. She's the one who decides the effective rank of A: singular values S(i) <= RCOND*S(1) get treated as zero, like they never even showed up 🙈

And if RCOND < 0, machine precision is used instead — the "I don't wanna pick, you pick" option fr fr.

RANK

RANK is INTEGER. The effective rank of A, i.e. the number of singular values which are greater than RCOND*S(1) ⭐

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, and she is the dimension of the array WORK. LWORK must be at least 1.

The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least

    2*N + N*NRHS

if M is greater than or equal to N, or

    2*M + M*NRHS

if M is less than N, the code will execute correctly ✅

For good performance, LWORK should generally be larger — she's not a "bare minimum" kind of girl, give her room to breathe 🦋

Ok so if LWORK = -1, a workspace query is assumed 😌 the routine only calculates the optimal size of the array WORK and the minimum sizes of the arrays RWORK and IWORK, and returns these values as the first entries of the WORK, RWORK and IWORK arrays — and no error message related to LWORK is issued by XERBLA. Like, she's just asking, don't yell at her 💅

RWORK

RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)). The requirements are:

    LRWORK >=
       10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
       MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )

if M is greater than or equal to N, or

    LRWORK >=
       10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
       MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )

if M is less than N, and then the code will execute correctly 😤

SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and

    NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK 🌈

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK)), and

    LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),

where MINMN = MIN( M,N ) 🧚

On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK 💜

INFO

INFO is INTEGER, the main character of the status report:

  • = 0: successful exit 🎉 slay
  • < 0: if INFO = -i, the i-th argument had an illegal value — rude of you honestly
  • > 0: the algorithm for computing the SVD failed to converge 😭 if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero

🌟 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Osni Marques, LBNL/NERSC, USA

Definition at line 217 of file zgelsd.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

okay that's the whole thing bestie, we ate and left no crumbs 👑 this little gem is the gelsd(3) page from the LAPACK source, Version 3.12.0 — go forth and solve those least-squares problems, ily, bye 🫶💖✨

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