SRC/cgelsd.f(3)

🌸 NAME 🌸

This is SRC/cgelsd.f from the Library Functions Manual — yes bestie, the whole page is about one iconic little subroutine. 💅

✨ SYNOPSIS ✨

🧩 Functions/Subroutines 🧩

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. slay. ✨

💖 Function/Subroutine Documentation 💖

Here’s the full signature, character-for-character, no cap:

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 🎯

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

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

It does this using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient — main character energy, honestly. 💜

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 the matrix A. M >= 0.
• N — N is INTEGER. The number of columns of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. 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, literally destroyed.
• LDA — LDA is INTEGER. The leading dimension of the array A. 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. 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)). 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)). 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.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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okay that’s the whole SRC/cgelsd.f page, bestie — LAPACK Version 3.12.0, SRC/cgelsd.f(3), and yes I’m still obsessed. 💅✨