complex16GEsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

✨ complex16GEsolve(3) ✨ — the complex16 GE solve-driver squad 💅🦋

🎀 NAME 🎀

complex16GEsolve — complex16

🌈 SYNOPSIS 🌈

💖 Functions 💖

  • subroutine zcgesv (N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, SWORK, RWORK, ITER, INFO) — ZCGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement)
  • subroutine zgels (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) — ZGELS solves overdetermined or underdetermined systems 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
  • subroutine zgelss (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, INFO) — ZGELSS solves overdetermined or underdetermined systems for GE matrices
  • subroutine zgelst (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) — ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.
  • subroutine zgelsy (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, RWORK, INFO) — ZGELSY solves overdetermined or underdetermined systems for GE matrices
  • subroutine zgesv (N, NRHS, A, LDA, IPIV, B, LDB, INFO) — ZGESV computes the solution to system of linear equations A * X = B for GE matrices (simple driver)
  • subroutine zgesvx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO) — ZGESVX computes the solution to system of linear equations A * X = B for GE matrices
  • subroutine zgesvxx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) — ZGESVXX computes the solution to system of linear equations A * X = B for GE matrices
  • subroutine zgetsls (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) — ZGETSLS

🌟 Detailed Description 🌟

This is the group of complex16 solve driver functions for GE matrices. Bestie, it's literally the whole clique. 💖

👑 Function Documentation 👑

🧚 subroutine zcgesv

subroutine zcgesv (integer N, integer NRHS, complex*16, dimension( lda, * )
A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension(
ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX,
complex*16, dimension( n, * ) WORK, complex, dimension( * ) SWORK,
double precision, dimension( * ) RWORK, integer ITER, integer INFO)

ZCGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement)

Purpose:

ZCGESV computes the solution to a complex system of linear equations

   A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

ZCGESV first attempts to factorize the matrix in COMPLEX and use this factorization within an iterative refinement procedure to produce a solution with COMPLEX*16 normwise backward error quality (see below). If the approach fails the method switches to a COMPLEX*16 factorization and solve.

The iterative refinement is not going to be a winning strategy if the ratio COMPLEX performance over COMPLEX*16 performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement. 🙈

The iterative refinement process is stopped if

   ITER > ITERMAX

or for all the RHS we have:

   RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where

  • ITER is the number of the current iteration in the iterative refinement process
  • RNRM is the infinity-norm of the residual
  • XNRM is the infinity-norm of the solution
  • ANRM is the infinity-operator-norm of the matrix A
  • EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.

Parameters

  • N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N coefficient matrix A. On exit, if iterative refinement has been successfully used (INFO = 0 and ITER >= 0, see description below), then A is unchanged; if double precision factorization has been used (INFO = 0 and ITER < 0, see description below), then the array A contains the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i). Corresponds either to the single precision factorization (if INFO = 0 and ITER >= 0) or the double precision factorization (if INFO = 0 and ITER < 0).
  • B — B is COMPLEX*16 array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — X is COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, the N-by-NRHS solution matrix X.
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • WORK — WORK is COMPLEX*16 array, dimension (N,NRHS). This array is used to hold the residual vectors.
  • SWORK — SWORK is COMPLEX array, dimension (N*(N+NRHS)). This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
  • RWORK — RWORK is DOUBLE PRECISION array, dimension (N)
  • ITER — ITER is INTEGER
    • < 0: iterative refinement has failed, COMPLEX*16 factorization has been performed
      • -1 : the routine fell back to full precision for implementation- or machine-specific reasons
      • -2 : narrowing the precision induced an overflow, the routine fell back to full precision
      • -3 : failure of CGETRF
      • -31: stop the iterative refinement after the 30th iterations
    • > 0: iterative refinement has been successfully used. Returns the number of iterations
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, U(i,i) computed in COMPLEX*16 is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

Author

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

🦋 subroutine zgels

subroutine zgels (character TRANS, integer M, integer N, integer NRHS,
complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer LWORK,
integer INFO)

ZGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

ZGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

The following options are provided:

  1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
                minimize || B - A*X ||.
  1. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.
  2. If TRANS = 'C' and m >= n: find the minimum norm solution of an underdetermined system A**H * X = B.
  3. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
                minimize || B - A**H * X ||.

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.

Parameters

  • TRANS — TRANS is CHARACTER*1
    • = 'N': the linear system involves A;
    • = 'C': the linear system involves A**H.
  • 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*16 array, dimension (LDA,N). On entry, the M-by-N matrix A.
    • if M >= N, A is overwritten by details of its QR factorization as returned by ZGEQRF;
    • if M < N, A is overwritten by details of its LQ factorization as returned by ZGELQF.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B —
  • B — it's a COMPLEX*16 array, dimension (LDB,NRHS). 💙

On entry, it's the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'C'.

On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise. And bestie, the cases are so specific, watch this:

  • if TRANS = 'N' and m >= n, rows 1 to n of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of the modulus of elements N+1 to M in that column;
  • if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors;
  • if TRANS = 'C' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
  • if TRANS = 'C' and m < n, rows 1 to M of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of the modulus of elements M+1 to N in that column.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= MAX(1,M,N). 📏
  • WORK — COMPLEX*16 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 >= max( 1, MN + max( MN, NRHS ) ).

For optimal performance, LWORK >= max( 1, MN + max( MN, NRHS )*NB ). where MN = min(M,N) and NB is the optimum block size. 🚀

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.

  • INFO — INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 😬
    • > 0: if INFO = i, the i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed. 💀

🫶 Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ 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 'bidiagonal least squares problem' (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.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. 🧚

📋 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 — COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, A has been destroyed. 💀 (gone, reduced to atoms)
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — COMPLEX*16 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,M,N).
  • S — 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 — DOUBLE PRECISION. 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 — COMPLEX*16 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
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 — DOUBLE PRECISION 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 — 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.
    • = 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

✨ subroutine zgelss ✨

subroutine zgelss (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 INFO)

ZGELSS solves overdetermined or underdetermined systems for GE matrices 💅

🎯 Purpose 🎯

ZGELSS computes the minimum norm solution to a complex 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 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 — COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise.
  • LDA — INTEGER, bestie. It's the leading dimension of the array A. Rule of the land: LDA >= max(1,M). no exceptions, periodt 💅
  • B — COMPLEX*16 array, dimension (LDB,NRHS). On entry it's the M-by-NRHS right hand side matrix B. On exit, B gets overwritten by the N-by-NRHS solution matrix X (she's a glow-up, not a copy ✨).

Also, important lil detail: 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. iykyk 🧮

  • LDB — INTEGER. Leading dimension of the array B. LDB >= max(1,M,N). don't skimp on her 🫶
  • S — 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)). she's giving main character energy, biggest first 👑
  • RCOND — DOUBLE PRECISION. RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero (like, ghosted 🫥). If RCOND < 0, machine precision is used instead. cute little fallback 💖
  • RANK — INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1). literally just counting the ones who survived the vibe check 🔥
  • WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she's a little spy, telling you what size you should've used 🙈
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= 1, and also:
    • LWORK >= 2*min(M,N) + max(M,N,NRHS)

For good performance, LWORK should generally be larger (bigger workspace = better performance, no cap 📈).

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. so go ahead, ask her, she won't yell at you 💅
  • RWORK — DOUBLE PRECISION array, dimension (5*min(M,N)). that's the whole entry, that's it, that's the tea 🌸
  • INFO — INTEGER. the receipts 📋
    • = 0: successful exit — we ate ✨
    • < 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. (she tried, she really did 😭)

Author 🎀

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

subroutine zgelst

subroutine zgelst (character TRANS, integer M, integer N, integer NRHS,
    complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
    ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer LWORK,
    integer INFO)

ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q. okay she's fancy 💅

Purpose 🦋

ZGELST solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A with compact WY representation of Q. It is assumed that A has full rank.

The following options are provided:

  1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
             minimize || B - A*X ||.
  1. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.
  2. If TRANS = 'C' and m >= n: find the minimum norm solution of an underdetermined system A**T * X = B.
  3. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
             minimize || B - A**T * X ||.

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. bulk mode activated, we love efficiency 🌈

Parameters 🌸

  • TRANS — CHARACTER*1
    • = 'N': the linear system involves A;
    • = 'C': the linear system involves A**H.
  • 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. (yes, zero space, keep it movin 🫶)
  • A — COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit:
    • if M >= N, A is overwritten by details of its QR factorization as returned by ZGEQRT;
    • if M < N, A is overwritten by details of its LQ factorization as returned by ZGELQT.

so she morphs depending on the shape of the situation, shapeshifter behavior 🦋

  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'C'. On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise:
    • if TRANS = 'N' and m >= n, rows 1 to n of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of modulus of elements N+1 to M in that column;
    • if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors;
    • if TRANS = 'C' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
    • if TRANS = 'C' and m < n, rows 1 to M of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of the modulus of elements M+1 to N in that column.

four whole moods, pick your fighter 🔥

  • LDB — INTEGER. The leading dimension of the array B. LDB >= MAX(1,M,N). uppercase MAX here, we respect her exactly as written 💜
  • WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. again with the little spoiler 📬
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= max( 1, MN + max( MN, NRHS ) ). For optimal performance, LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ). where MN = min(M,N) and NB is the optimum block size.
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. ask away bestie, no XERBLA screaming this time 💅
  • INFO — INTEGER
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed. 💀

Author 🎀

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

Contributors 🌟

  • November 2022, Igor Kozachenko, Computer Science Division, University of California, Berkeley

subroutine zgelsy

subroutine zgelsy (integer M, integer N, integer NRHS, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
    integer LDB, integer, dimension( * ) JPVT, double precision RCOND,
    integer RANK, complex*16, dimension( * ) WORK, integer LWORK, double
    precision, dimension( * ) RWORK, integer INFO)

ZGELSY solves overdetermined or underdetermined systems for GE matrices — yes girl, another one, and she's the rank-deficient-friendly icon 👑

Purpose 💖

ZGELSY computes the minimum-norm solution to a complex linear least squares problem:

    minimize || A * X - B ||

using a complete orthogonal factorization of A. A is an M-by-N matrix which may be rank-deficient. (finally, a routine that doesn't judge 🥹)

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 routine first computes a QR factorization with column pivoting:

    A * P = Q * [ R11 R12 ]
                [  0  R22 ]

with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/RCOND. The order of R11, RANK, is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated by unitary transformations from the right, arriving at the complete orthogonal factorization:

   A * P = Q * [ T11 0 ] * Z
               [  0  0 ]

The minimum-norm solution is then

   X = P * Z**H [ inv(T11)*Q1**H*B ]
                [        0         ]

where Q1 consists of the first RANK columns of Q. cute little pipeline, obsessed 🌈

This routine is basically identical to the original xGELSX except three differences:

  • The permutation of matrix B (the right hand side) is faster and more simple.
  • The call to the subroutine xGEQPF has been substituted by the the call to the subroutine xGEQP3. This subroutine is a Blas-3 version of the QR factorization with column pivoting.
  • Matrix B (the right hand side) is updated with Blas-3.

(and yes, "the the" is right there in the source, we reproduce her faithfully 🫶)

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 matrices B and X. NRHS >= 0.
  • A — COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, A has been overwritten by details of its complete orthogonal factorization. (she went in one girl and came out a whole factorization 💅)
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B
  • B — COMPLEX*16 array, dimension (LDB,NRHS). On the way in, this is your M-by-NRHS right hand side matrix B. On the way out? It's the N-by-NRHS solution matrix X. glow-up fr 💅
  • LDB — INTEGER, bestie. The leading dimension of the array B. And it's gotta be LDB >= max(1,M,N). no exceptions, she's strict 💀
  • JPVT — INTEGER array, dimension (N). On entry: if JPVT(i) .ne. 0, then the i-th column of A gets permuted to the front of AP, otherwise column i is a free column 🕊️. On exit: if JPVT(i) = k, then the i-th column of A*P was the k-th column of A. it's giving "where did everyone go" energy ✨
  • RCOND — DOUBLE PRECISION. RCOND is what we use to figure out the effective rank of A, which is defined as the order of the largest leading triangular submatrix R11 in the QR factorization with pivoting of A, whose estimated condition number < 1/RCOND. math girlies stay winning 🎀
  • RANK — INTEGER. The effective rank of A, i.e., the order of the submatrix R11. And this is literally the same as the order of the submatrix T11 in the complete orthogonal factorization of A. same energy, different name 👑
  • WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, then WORK(1) returns the optimal LWORK. she's the main character of the workspace 🫶
  • LWORK — INTEGER. The dimension of the array WORK. okay so listen up, because this one's got lore:

The unblocked strategy requires that:

LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )

where MN = min(M,N).

The block algorithm requires that:

LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )

where NB is an upper bound on the blocksize returned by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ. that's a whole squad of routines, iconic honestly 🌈

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. so don't come for XERBLA, she's off duty 💀

  • RWORK — DOUBLE PRECISION array, dimension (2*N). short and sweet, we love her 🍓
  • INFO — INTEGER. The vibe check of the whole operation:
    • = 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 💖

  • A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
  • E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
  • G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

✨ subroutine zgesv ✨

   subroutine zgesv (integer N, integer NRHS, complex*16, dimension( lda, * )
       A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension(
       ldb, * ) B, integer LDB, integer INFO)

ZGESV computes the solution to system of linear equations A * X = B for GE matrices (simple driver). she's the lowkey one, no scaling drama, just vibes and LU 🔥

🌸 Purpose 🌸

ZGESV computes the solution to a complex system of linear equations

   A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as

   A = P * L * U,

where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations A * X = B. periodt ✨

🎀 Parameters 🎀

  • N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. the more the merrier bestie 💅
  • A — COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N coefficient matrix A. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored. she doesn't need to be written down to be real 🫶
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • IPIV — INTEGER array, dimension (N). The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i). musical chairs but make it linear algebra 🪑✨
  • B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the N-by-NRHS matrix of right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. glow-up complete 💖
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INTEGER:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 🙈
    • > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. it's giving "we tried bestie" 😭

👑 Author 👑

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

✨ subroutine zgesvx ✨

   subroutine zgesvx (character FACT, character TRANS, integer N, integer NRHS,
       complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
       ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character
       EQUED, double precision, dimension( * ) R, double precision, dimension(
       * ) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16,
       dimension( ldx, * ) X, integer LDX, double precision RCOND, double
       precision, dimension( * ) FERR, double precision, dimension( * ) BERR,
       complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK,
       integer INFO)

ZGESVX computes the solution to system of linear equations A * X = B for GE matrices. she's the expert driver — error bounds, condition estimates, the whole production 🔥

🌸 Purpose 🌸

ZGESVX uses the LU factorization to compute the solution to a complex system of linear equations

   A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. she does it all, no cap 💅

🦋 Description 🦋

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C'). balance is key bestie ⚖️

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

  1. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. not me warning you and then finishing the job anyway 😌💅
  2. The system of equations is solved for X using the factored form of A.
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. she said "let me perfect that" ✨
  4. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration. putting it back the way we found it, we're respectful 🫶

🎀 Parameters 🎀

  • FACT — CHARACTER*1. Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.
    • = 'F': On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. A, AF, and IPIV are not modified.
    • = 'N': The matrix A will be copied to AF and factored.
    • = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
  • TRANS — CHARACTER*1. Specifies the form of the system of equations:
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Conjugate transpose)
  • N — INTEGER. The number of linear equations, i.e., the order 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 — COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

  • EQUED = 'R': A := diag(R) * A
  • EQUED = 'C': A := A * diag(C)
  • EQUED = 'B': A := diag(R) * A * diag(C).
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — COMPLEX*16 array, dimension (LDAF,N). If FACT = 'F', then AF is an input argument and on entry contains the factors L and U from the factorization A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A.

If FACT = 'N', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the original matrix A.

  • If FACT = 'E', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). 💫
  • LDAF — INTEGER. The leading dimension of the array AF. Constraint time: LDAF >= max(1,N). Don't play with this one bestie, she's strict. ✋
  • IPIV — INTEGER array, dimension (N). The pivot indices, and her whole personality depends on FACT:
    • If FACT = 'F', then IPIV is an input argument, and on entry it holds the pivot indices from the factorization A = P*L*U as computed by ZGETRF; row i of the matrix was interchanged with row IPIV(i).
    • If FACT = 'N', then IPIV is an output argument, and on exit it holds the pivot indices from the factorization A = P*L*U of the original matrix A.
    • If FACT = 'E', then IPIV is an output argument, and on exit it holds the pivot indices from the factorization A = P*L*U of the equilibrated matrix A.
  • EQUED — CHARACTER*1. She tells you which flavor of equilibration went down:
    • = 'N': No equilibration (this is always true if FACT = 'N').
    • = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
    • = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
    • = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).

EQUED is an input argument if FACT = 'F'; otherwise, it's an output argument. She switches sides depending on the situation, lowkey iconic. 💅

  • R — DOUBLE PRECISION array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed (she's off the clock, leave her alone 🙈). R is an input argument if FACT = 'F'; otherwise, R is an output argument. And heads up: if FACT = 'F' and EQUED = 'R' or 'B', every single element of R must be positive. No zeros, no negatives, periodt. ⚠️
  • C — DOUBLE PRECISION array, dimension (N). The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argument if FACT = 'F'; otherwise, C is an output argument. Same rule as her sister R: if FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. 🫶
  • B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, she's the N-by-NRHS right hand side matrix B. On exit it depends on the vibes:
    • if EQUED = 'N', B is not modified;
    • if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
    • if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
  • LDB — INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,N). 📏
  • X — COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X to the original system of equations. Real talk though: note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. So don't come for her if you skipped the equilibration math, bestie. 😭
  • LDX — INTEGER. The leading dimension of the array X. Constraint: LDX >= max(1,N). ✨
  • RCOND — DOUBLE PRECISION. The estimate of the reciprocal condition number of the matrix A after equilibration (if equilibration happened). If RCOND is less than the machine precision (and especially if RCOND = 0), the matrix is singular to working precision. That situation gets flagged by a return code of INFO > 0. 🚨
  • FERR — DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), then FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. She's cautious, not wrong. 💖
  • BERR — DOUBLE PRECISION array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
  • WORK — COMPLEX*16 array, dimension (2N)*. She's the scratch space. No notes. 🎀
  • RWORK — DOUBLE PRECISION array, dimension (2N)*. On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used here. If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, the condition estimator RCOND, and the forward error bound FERR could all be unreliable. And if factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. ⚠️ Not a drill.
  • INFO — INTEGER. The receipts:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, and i is
      • <= N: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
      • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed, because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. Plot twist, she came through anyway. 👑

💖 Author 💖

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

🌈 subroutine zgesvxx

subroutine zgesvxx (character FACT, character TRANS, integer N, integer
    NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16,
    dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
    character EQUED, double precision, dimension( * ) R, double precision,
    dimension( * ) C, complex*16, dimension( ldb, * ) B, integer LDB,
    complex*16, dimension( ldx , * ) X, integer LDX, double precision RCOND,
    double precision RPVGRW, double precision, dimension( * ) BERR, integer
    N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double
    precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double
    precision, dimension( * ) PARAMS, complex*16, dimension( * ) WORK,
    double precision, dimension( * ) RWORK, integer INFO)

ZGESVXX computes the solution to system of linear equations A * X = B for GE matrices

🦋 Purpose 🦋

ZGESVXX uses the LU factorization to compute the solution to a complex*16 system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds are returned. ZGESVXX will return a solution with a tiny guaranteed error (O(eps) where eps is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers also are calculated and returned. She's thorough like that. 💅

ZGESVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous ZGESVXX call will also produce a solution with either O(eps) errors or warnings — but we cannot make that claim for general user-provided factorizations and equilibration factors if they differ from what ZGESVXX would itself produce. So like, if you bring your own factorization and it's different from hers, that's on you bestie. 🙈

📋 Description 📋

The following steps are performed:

  1. If FACT = 'E', double precision scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

  1. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A (see argument RCOND). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for X and compute error bounds as described below. She does not give up. 🫶
  2. The system of equations is solved for X using the factored form of A.
  3. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), the routine will use iterative refinement to try to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision. Fancy. ✨
  4. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration.

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but honestly? the defaults are usually serving. If the defaults are acceptable, users can pass NPARAMS = 0, which keeps the source code from ever touching the PARAMS argument. 💅

🎀 Parameters 🎀

  • FACT — FACT is CHARACTER*1. It tells the routine whether the factored form of the matrix A is being handed over on entry, and if it's not, whether A should get equilibrated before it's factored. The tea:
    • = 'F': On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. A, AF, and IPIV are not modified.
    • = 'N': The matrix A will be copied to AF and factored.
    • = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
  • TRANS — TRANS is CHARACTER*1. Bestie, this is literally just which form of the system of equations you want:
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Conjugate Transpose)
  • N — N is INTEGER. The number of linear equations, aka the order of the matrix A. So N >= 0. ✨
  • NRHS — NRHS is INTEGER. The number of right hand sides, aka the number of columns of the matrices B and X. NRHS >= 0.
  • A — A is COMPLEX*16 array, dimension (LDA,N). On entry, this is the N-by-N matrix A. If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. A stays untouched if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled like this:

EQUED = 'R':  A := diag(R) * A
EQUED = 'C':  A := A * diag(C)
EQUED = 'B':  A := diag(R) * A * diag(C).
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is COMPLEX*16 array, dimension (LDAF,N). Depends on the FACT vibe:
    • If FACT = 'F', then AF is an input argument, and on entry it contains the factors L and U from the factorization A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A.
    • If FACT = 'N', then AF is an output argument, and on exit it returns the factors L and U from the factorization A = P*L*U of the original matrix A.
    • If FACT = 'E', then AF is an output argument, and on exit it returns the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (peep the description of A for what that equilibrated matrix even looks like).
  • LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). Again, FACT decides everything:
    • If FACT = 'F', then IPIV is an input argument, and on entry it contains the pivot indices from the factorization A = P*L*U as computed by ZGETRF; row i of the matrix was interchanged with row IPIV(i).
    • If FACT = 'N', then IPIV is an output argument, and on exit it contains the pivot indices from the factorization A = P*L*U of the original matrix A.
    • If FACT = 'E', then IPIV is an output argument, and on exit it contains the pivot indices from the factorization A = P*L*U of the equilibrated matrix A.
  • EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done:
    • = 'N': No equilibration (always true if FACT = 'N').
    • = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
    • = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
    • = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.

  • R — R is DOUBLE PRECISION array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. If R is output, each element of R is a power of the radix. If R is input, each element of R should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
  • C — C is DOUBLE PRECISION array, dimension (N). The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. If C is output, each element of C is a power of the radix. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
  • B — B is COMPLEX*16 array, dimension (LDB,NRHS). On entry, it's the N-by-NRHS right hand side matrix B. On exit:
    • if EQUED = 'N', B is not modified;
    • if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
    • if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — X is COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. 💖
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — RCOND is DOUBLE PRECISION. Reciprocal scaled condition number. It's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that happened). If this is less than the machine precision (in particular, if it's zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. So don't panic-panic. 🫶
  • RPVGRW — RPVGRW is DOUBLE PRECISION. Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. In ZGESVX, this quantity is returned in WORK(1).
  • BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
  • N_ERR_BNDS — N_ERR_BNDS is INTEGER. Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. 👑
  • ERR_BNDS_NORM — ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon').
  • err = 2 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — Reciprocal condition number vibes: this is the estimated normwise reciprocal condition number. It gets compared against the threshold sqrt(n) * dlamch('Epsilon') to figure out whether the error estimate is 'guaranteed' or not. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Like, let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. ✨

See Lapack Working Note 165 for further details and extra cautions, bestie — no cap, there's more lore in there.

✨ ERR_BNDS_COMP ✨

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array is literally giving you info about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined like this:

Componentwise relative error in the ith solution vector:

               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends, bestie, it's giving main character energy), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed at all — it's giving ignored 💅. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. 🎀

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). trust issues? valid.
  • err = 2 — 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 🫶
  • err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. 🦋

See Lapack Working Note 165 for further details and extra cautions.

✨ NPARAMS ✨

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used instead. respecting the boundaries, we love to see it.

✨ PARAMS ✨

PARAMS is DOUBLE PRECISION array, dimension NPARAMS

Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. 💖

  • PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not.
    • Default: 1.0D+0
      • = 0.0: No refinement is performed, and no error bounds are computed.
      • = 1.0: Use the extra-precise refinement algorithm.
      • (other values are reserved for future use, bestie — don't try it)
  • PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement.
    • Default: 10
    • Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. 😭
  • PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
    • Default: 1.0 (attempt componentwise convergence) 🔥

✨ WORK ✨

WORK is COMPLEX*16 array, dimension (2*N)

✨ RWORK ✨

RWORK is DOUBLE PRECISION array, dimension (2*N)

✨ INFO ✨

INFO is INTEGER
  • = 0: Successful exit. The solution to every right-hand side is guaranteed. slay! ✅
  • < 0: If INFO = -i, the i-th argument had an illegal value. oops.
  • > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
  • = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. 🧚

Author

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

🌈 subroutine zgetsls 🌈

subroutine zgetsls (character TRANS, integer M, integer N, integer NRHS,
    complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
    ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer LWORK,
    integer INFO)

ZGETSLS

Purpose:

ZGETSLS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, using a tall skinny QR or short wide LQ factorization of A. It is assumed that A has full rank. the assumption is giving unbothered, don't break it.

The following options are provided:

  1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
            minimize || B - A*X ||.
  1. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.
  2. If TRANS = 'C' and m >= n: find the minimum norm solution of an undetermined system A**T * X = B.
  3. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
            minimize || B - A**T * X ||.

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. 🎀

Parameters

TRANS

TRANS is CHARACTER*1
  • = 'N': the linear system involves A;
  • = 'C': the linear system involves A**H.

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*16 array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, A is overwritten by details of its QR or LQ factorization as returned by ZGEQR or ZGELQ. 💅

LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

B

B is COMPLEX*16 array, dimension (LDB,NRHS)

On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'C'.

On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise:

  • if TRANS = 'N' and m >= n, rows 1 to n of B contain the least squares solution vectors.
  • if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors;
  • if TRANS = 'C' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
  • if TRANS = 'C' and m < n, rows 1 to M of B contain the least squares solution vectors.

LDB

LDB is INTEGER

The leading dimension of the array B. LDB >= MAX(1,M,N).

WORK

(workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details. ✨

LWORK

LWORK is INTEGER

The dimension of the array WORK.

If LWORK = -1 or -2, then a workspace query is assumed.

If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1).

If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1).

INFO

  • INFO is an INTEGER (yes bestie, it's an out-param, it tells you what happened) 💅
    • = 0 — successful exit. We love a zero. No notes. ✨
    • < 0 — if INFO = -i, then the i-th argument had an illegal value 🙈 you passed something cursed, go check argument number i.
    • > 0 — if INFO = i, then the i-th diagonal element of the triangular factor of A is zero, so A does not have full rank; the least squares solution could not be computed 😭 that's the whole point of the call, so… yeah. Not great.

👑 Author

Like, the actual authors, the ones who did the whole thing:

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

🖋️ Author (again, but make it generated)

Generated automatically by Doxygen for LAPACK from the source code. Not a human typing this by hand, no cap — the robot did it, and honestly? It ate. 🤖✨

fr fr, that's the whole page. Every option, every INFO value, every author, all of it — preserved, just with a little more ✨vibe✨. The way that this documentation is giving "written by a committee of very smart people and then blessed by a Doxygen script" 🎀

Ok that's it, we're done, the manpage said its thing 💖 This one came from LAPACK, Version 3.11.0, dated Sun Nov 27 2022, and complex16GEsolve(3) is officially wrapped. Slay, go solve some least squares problems (but only if A has full rank, otherwise… you know) 🦋🌸

📦 where this page comes from