doubleGEsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

👑 NAME 👑

doubleGEsolve — double

okay so real talk, bestie: this is the LAPACK manual, section 3 (that's the library-routines section, iykyk), and this page is the whole lil squad of double-precision GE solve drivers. We're talking all the routines that take a general matrix A and go "say less, I'll get you your solution vectors." 💅

✨ SYNOPSIS ✨

🦋 Functions 🦋

  • subroutine dgels (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) — DGELS solves overdetermined or underdetermined systems 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 dgelss (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO) — DGELSS solves overdetermined or underdetermined systems for GE matrices
  • subroutine dgelst (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) — DGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.
  • subroutine dgelsy (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO) — DGELSY solves overdetermined or underdetermined systems for GE matrices
  • subroutine dgesv (N, NRHS, A, LDA, IPIV, B, LDB, INFO) — DGESV computes the solution to system of linear equations A * X = B for GE matrices
  • subroutine dgesvx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) — DGESVX computes the solution to system of linear equations A * X = B for GE matrices
  • subroutine dgesvxx (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, IWORK, INFO) — DGESVXX computes the solution to system of linear equations A * X = B for GE matrices
  • subroutine dgetsls (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) — DGETSLS
  • subroutine dsgesv (N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, SWORK, ITER, INFO) — DSGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement)

🌈 Detailed Description 🌈

This is the group of double solve driver functions for GE matrices. That's it, that's the whole vibe. 🫶

💖 Function Documentation 💖

🎀 subroutine dgels

subroutine dgels (character TRANS, integer M, integer N, integer NRHS,
    double precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldb, * ) B, integer LDB, double precision, dimension( * )
    WORK, integer LWORK, integer INFO)

DGELS solves overdetermined or underdetermined systems for GE matrices — she's the main character of this whole page, ngl. 👑

🎯 Purpose:

DGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its 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 = 'T' and m >= n: find the minimum norm solution of an underdetermined system A**T * X = B.
  3. If TRANS = 'T' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
             minimize || B - A**T * X ||.

And the glow-up? Several right hand side vectors b and solution vectors x can be handled in a single call 🫶 — they're stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. One call, many vectors, we love efficiency. ✨

🌸 Parameters

  • TRANS
    • TRANS is CHARACTER*1
    • = 'N': the linear system involves A;
    • = 'T': the linear system involves A**T.
  • 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 DOUBLE PRECISION 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 DGEQRF;
      • if M < N, A is overwritten by details of its LQ factorization as returned by DGELQF.
  • LDA
    • LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,M).
  • B
    • B is DOUBLE PRECISION 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 = 'T'.
    • 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 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 = 'T' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
      • if TRANS = 'T' 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 elements M+1 to N in that column.
  • LDB
    • LDB is INTEGER
    • The leading dimension of the array B. LDB >= MAX(1,M,N).
  • WORK
    • WORK is DOUBLE PRECISION 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 >= 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. (sneaky lil shortcut, she ate 💅)
  • INFO
    • INFO is 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.
ngl bestie, watch that INFO > 0 case — no full rank means no least squares solution for you. She's not being dramatic, it literally could not be computed. 😭

🫶 Author

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

🦋 subroutine 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 — and she does it with the SVD, so you know she's fancy. ✨

🎯 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 '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. Lowkey iconic that they just say "we know of none" 💀

🌸 Parameters

  • M
    • M is INTEGER
    • The number of rows of A. M >= 0.
  • N
    • N is INTEGER
    • The number of columns of 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 DOUBLE PRECISION array, dimension (LDA,N)
    • On entry, the M-by-N matrix A.
    • On exit, A has been destroyed. (yes girl, gone, reduced to atoms 🙈)
  • LDA
    • LDA is INTEGER
    • The leading dimension of the array A. 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. 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
    • RCOND is 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 — RANK is INTEGER, and it's the effective rank of A, i.e. the number of singular values that are greater than RCOND*S(1). that's the whole vibe check, no cap 💅
  • WORK — WORK is a DOUBLE PRECISION array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK for you bestie ✨
  • LWORK — LWORK is INTEGER, and it's 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. fr fr, do not be stingy with the workspace 🙈

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.

  • IWORK — IWORK is an 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, and it's literally the receipts:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. rude, honestly.
    • > 0: the algorithm for computing the SVD failed to converge; if INFO = i, then 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 dgelss ✨

subroutine dgelss (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 INFO)

DGELSS solves overdetermined or underdetermined systems for GE matrices 🦋

💖 Purpose 💖

DGELSS 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 — yes, it's allowed to be a little messy, we still love it.

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. so many problems solved at once, she's giving main character energy 👑

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 a DOUBLE PRECISION 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. she got a glow-up on the way out, obsessed 💅
  • LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,M).
  • B — B is a 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. LDB >= max(1,max(M,N)).
  • S — S is a 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)). the numbers go down, never up ✨
  • RCOND — RCOND is 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 — 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 a DOUBLE PRECISION 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 >= 1, and also:
LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )

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.

  • 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, then 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.

🌈 subroutine dgelst 🌈

subroutine dgelst (character TRANS, integer M, integer N, integer NRHS,
    double precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldb, * ) B, integer LDB, double precision, dimension( * )
    WORK, integer LWORK, integer INFO)

DGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q. the way that it does QR and LQ... versatile icon 🧚

💖 Purpose 💖

DGELST solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A with compact WY representation of Q. it is assumed that A has full rank — no rank-deficiency chaos here bestie, this one wants a clean slate.

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 = 'T' and m >= n: find the minimum norm solution of an underdetermined system A**T * X = B.
  3. If TRANS = 'T' 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 processing, we love to see it 🍓

🎀 Parameters 🎀

  • TRANS — TRANS is CHARACTER*1
    • = 'N': the linear system involves A;
    • = 'T': the linear system involves A**T.
  • 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 a DOUBLE PRECISION 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 DGEQRT;
    • if M < N, A is overwritten by details of its LQ factorization as returned by DGELQT.
  • LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,M).
  • B — B is a DOUBLE PRECISION 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 = 'T'. 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 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 = 'T' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
    • if TRANS = 'T' 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 elements M+1 to N in that column.
  • LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= MAX(1,M,N).
  • WORK — WORK is a DOUBLE PRECISION array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK ⭐
  • LWORK —
  • LWORK (INTEGER) — bestie, this is just how big the WORK array is. And she's gotta be big enough, no cap:
LWORK >= max( 1, MN + max( MN, NRHS ) ).

For optimal performance (aka the glow-up version), you want:

LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).

where MN = min(M,N) and NB is the optimum block size. ✨

🫶 Special shortcut: if LWORK = -1, it's a workspace query moment — the routine doesn't actually do the work, it just figures out the optimal size of the WORK array and hands it back as the first entry of WORK. And no error message related to LWORK gets thrown by XERBLA. It's giving "just checking, not committing" energy. 💅
  • INFO (INTEGER) — the receipts, literally:
    • = 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 dgelsy 💖

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

DGELSY solves overdetermined or underdetermined systems for GE matrices — she does it all, honestly.

🦋 Purpose: 🦋

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

minimize || A * X - B ||

and she does it using a complete orthogonal factorization of A. Now, A is an M-by-N matrix which may be rank-deficient — and that's fine, she's not judging.

Several right hand side vectors b and solution vectors x can be handled in a single call (efficient queen behavior ✨); 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 (we don't claim it, we ignore it 💅), and R12 is annihilated by orthogonal 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**T [ inv(T11)*Q1**T*B ]
             [        0         ]

where Q1 consists of the first RANK columns of Q.

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

  • 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.
  • The permutation of matrix B (the right hand side) is faster and more simple. ✨

🌸 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 (DOUBLE PRECISION 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 took over, bestie. 💅
  • LDA (INTEGER) — the leading dimension of the array A. LDA >= max(1,M).
  • B (DOUBLE PRECISION array, dimension (LDB,NRHS)) — on entry, the M-by-NRHS right hand side matrix B. On exit, the N-by-NRHS solution matrix X. Glow up ✨
  • LDB (INTEGER) — the leading dimension of the array B. LDB >= max(1,M,N).
  • JPVT (INTEGER array, dimension (N)) — on entry, if JPVT(i) .ne. 0, the i-th column of A is 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 AP was the k-th column of A.
  • RCOND (DOUBLE PRECISION) — RCOND is used to determine 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.
  • RANK (INTEGER) — the effective rank of A, i.e., the order of the submatrix R11. This is the same as the order of the submatrix T11 in the complete orthogonal factorization of A.
  • WORK (DOUBLE PRECISION array, dimension (MAX(1,LWORK))) — on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK (INTEGER) — the dimension of the array WORK.

The unblocked strategy requires that:

LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),

where MN = min( M, N ).

The block algorithm requires that:

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

where NB is an upper bound on the blocksize returned by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR, and DORMRZ.

🫶 And yes, 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. Same little trick as before, iconic.
  • INFO (INTEGER) — the vibes report:
    • = 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 dgesv 💜

subroutine dgesv (integer N, integer NRHS, double precision, dimension( lda,
    * ) A, integer LDA, integer, dimension( * ) IPIV, double precision,
    dimension( ldb, * ) B, integer LDB, integer INFO)

DGESV computes the solution to system of linear equations A * X = B for GE matrices — the classic, the blueprint, the main character. 🌟

🦋 Purpose: 🦋

DGESV computes the solution to a real 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. Slay. ✨

🌸 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.
  • A (DOUBLE PRECISION 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. (Not me storing extra stuff I don't need 🙈)
  • 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).
  • B (DOUBLE PRECISION 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. ✨
  • LDB (INTEGER) — the leading dimension of the array B. LDB >= max(1,N).
  • INFO (INTEGER) — receipts time:
    • = 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. 💀

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 subroutine dgesvx 🧚

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

DGESVX computes the solution to system of linear equations A * X = B for GE matrices — but make it ✨extra✨.

🦋 Purpose: 🦋

DGESVX uses the LU factorization to compute the solution to a real 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's not just giving you the answer, she's giving you the confidence interval. 👑

🍓 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'). ✨

  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. 🫶
  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.
  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.

✨ Parameters ✨

  • FACT — FACT is 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 — TRANS is 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 (Transpose)
  • 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 matrices B and X. NRHS >= 0.
  • A — A is DOUBLE PRECISION 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is DOUBLE PRECISION 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 DGETRF. 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 — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A = P*L*U as computed by DGETRF; row i of the matrix was interchanged with row IPIV(i).
    • If FACT = 'N', then IPIV is an output argument and on exit 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 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. 🔥
  • 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. 🔥
  • B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, 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 DOUBLE PRECISION array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, 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. The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. 💀
  • FERR — FERR is 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), 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 is almost always a slight overestimate of the true error.
  • BERR — BERR is 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 — WORK is DOUBLE PRECISION array, dimension (4*N). On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If WORK(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, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then WORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. 😭
  • IWORK — IWORK is INTEGER array, dimension (N)
  • INFO —
  • = 0 — successful exit, we ate 💅
  • < 0 — if INFO = -i, bestie, the i-th argument had an illegal value. not slay.
  • > 0 — if INFO = i, and i is one of these little guys:
    • <= 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. it's giving broken 💀
    • = N+1 — U is nonsingular, but RCOND is less than machine precision, meaning 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, the girlies pulled through 😭

🎀 Author 🎀

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine dgesvxx 🦋

subroutine dgesvxx (character FACT, character TRANS, integer N, integer
    NRHS, double precision, dimension( lda, * ) A, integer LDA, double
    precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( *
    ) IPIV, character EQUED, double precision, dimension( * ) R, double
    precision, dimension( * ) C, double precision, dimension( ldb, * ) B,
    integer LDB, double precision, 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,
    double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,
    integer INFO)

DGESVXX computes the solution to system of linear equations A * X = B for GE matrices. no cap ✨

💖 Purpose 💖

DGESVXX uses the LU factorization to compute the solution to a double precision 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. DGESVXX 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. like she really does the most 🫶

DGESVXX 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 DGESVXX 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 DGESVXX would itself produce. read that twice 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's resilient like that 💪
  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. extra, but make it accurate ✨
  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 often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. lazy girl hack, approved 💅

⭐ Parameters ⭐

FACT

FACT is 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

TRANS is 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 = Transpose)

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 matrices B and X. NRHS >= 0.

A

A is DOUBLE PRECISION 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

LDA is INTEGER

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

AF

AF is DOUBLE PRECISION 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 DGETRF. 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

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N)

If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A = P*L*U as computed by DGETRF; row i of the matrix was interchanged with row IPIV(i).

If FACT = 'N', then IPIV is an output argument and on exit 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 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. 🫣

bestie, that last bit is the real tea — if your scale factors aren't powers of the radix, your error estimates are lowkey lying to you.

C

  • C — DOUBLE PRECISION array, dimension (N). These are the column scale factors for A 💜. If EQUED = 'C' or 'B', then A gets multiplied on the right by diag(C); if EQUED = 'N' or 'R', bestie, C is literally not accessed at all — she's just sitting there unbothered. C is an input argument if FACT = 'F'; otherwise C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', every single 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, because that's what gets you a reliable solution and reliable 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, which produces error estimates that may not be reliable 🫠
  • B — DOUBLE PRECISION array, dimension (LDB,NRHS).
    • On entry, 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,N). That's the vibe check, no exceptions 👑
  • X — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — DOUBLE PRECISION. Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is 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. Iconic little plot twist honestly.
  • RPVGRW — 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, the estimated condition numbers, and the 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 DGESVX, this quantity is returned in WORK(1).
  • BERR — 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 — 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 — 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: estimated normwise 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, 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 📖

  • ERR_BNDS_COMP — 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 componentwise relative error, which is defined as follows:
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), 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. 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').
  • 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 — INTEGER. Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Respectfully, she's not even looking at it 🫶
  • PARAMS — 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)
    • 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 — DOUBLE PRECISION array, dimension (4*N).
  • IWORK — INTEGER array, dimension (N).
  • INFO
  • INFO = 0 — everything went off without a hitch, the exit is successful, and bestie, the solution for every single right-hand side is guaranteed. 💯
  • INFO < 0 — if INFO = -i, then the i-th argument was giving illegal value energy, and we simply cannot. 🚫
  • INFO > 0 and <= N — so U(INFO,INFO) is exactly zero. The factorization finished its whole thing, but the factor U is exactly singular, which means the solution and the error bounds could not be computed at all. RCOND = 0 gets returned. It's giving 💀, honestly.
  • INFO = N+J — the solution for the Jth right-hand side is not guaranteed. And the solutions for the other right-hand sides K with K > J may not be guaranteed either, but only the first such right-hand side gets reported. So here's the tea: if a small componentwise error was not requested (PARAMS(3) = 0.0), then the Jth right-hand side is the first one with a normwise error bound that is not guaranteed — that's the smallest J such that ERR_BNDS_NORM(J,1) = 0.0. But by default (PARAMS(3) = 1.0), the Jth right-hand side is the first one with either a normwise or a componentwise error bound that isn't guaranteed — the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0. Go look at the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1) for the deets. And if you wanna know about all of the right-hand sides, check ERR_BNDS_NORM or ERR_BNDS_COMP. periodt. 💅

👑 Author 👑

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

✨ dgetsls ✨

subroutine dgetsls (character TRANS, integer M, integer N, integer NRHS,
    double precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldb, * ) B, integer LDB, double precision, dimension( * )
    WORK, integer LWORK, integer INFO)

DGETSLS 🦋

🌸 Purpose 🌸

DGETSLS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, using a tall skinny QR or a short wide LQ factorization of A. And it just assumes A has full rank, no questions asked. ✨

The options on the menu, bestie:

  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 ||.
  2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.
  3. If TRANS = 'T' and m >= n: find the minimum norm solution of an undetermined system A**T * X = B.
  4. If TRANS = 'T' 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 get stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. one call, so many solutions, we love efficiency. 🔥

🎀 Parameters 🎀

  • TRANS — TRANS is CHARACTER*1
    • = 'N': the linear system involves A;
    • = 'T': the linear system involves A**T.
  • 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 DOUBLE PRECISION 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 DGEQR or DGELQ.
  • LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
  • B — B is DOUBLE PRECISION 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 = 'T'. 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 = 'T' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
    • if TRANS = 'T' 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) DOUBLE PRECISION 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 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.

🧚 dsgesv 🧚

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

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

⭐ Purpose ⭐

DSGESV computes the solution to a real system of linear equations

A * X = B,

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

So here's the strategy, bestie: DSGESV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below, iykyk). And if that approach fails, the method switches to a DOUBLE PRECISION factorization and solve. it's giving backup plan energy. 🙈

The iterative refinement is not going to be a winning strategy if the ratio of SINGLE PRECISION performance over DOUBLE PRECISION 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 — but up to now, we always try iterative refinement. no cap. 💅

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. locked in. 🔒

🍓 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N,NRHS) This array is used to hold the residual vectors.
  • SWORK — SWORK is REAL 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.

ITER is INTEGER, and bestie, this is the whole story of how the iterative refinement went. No cap:

  • < 0 — iterative refinement has failed, double precision 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 SGETRF
    • -31 — stop the iterative refinement after the 30th iterations
  • > 0 — iterative refinement has been successfully used. Returns the number of iterations

INFO is INTEGER — the main character of exit statuses:

  • = 0 — successful exit
  • < 0 — if INFO = -i, the i-th argument had an illegal value
  • > 0 — if INFO = i, U(i,i) computed in DOUBLE PRECISION is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.
🚨 if INFO > 0, bestie, the factorization finished but U is exactly singular, so the solution could not be computed. That's not slay, that's a whole crisis. 💀

Author 💖

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

Author 🌸

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✨ Source/version check: LAPACK 3.11.0, dated Sun Nov 27 2022 — and this whole page is doubleGEsolve(3). Periodt. 💅🫶
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