SRC/dgesvx.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

🎀 SRC/dgesvx.f 🎀

okay bestie, buckle up — this is dgesvx from the LAPACK gang, and she is that girl. Manual page name: SRC/dgesvx.f(3), library functions manual, and yes she's iconic. 💅

✨ SYNOPSIS ✨

🌸 Functions/Subroutines

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 — literally her whole personality.

🦋 Function/Subroutine Documentation 🦋

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 🫶 (yes that's the one-liner again, she said it twice, she's confident)

💖 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.
!>

so real talk: she solves A * X = B for a real system, A is N-by-N, and X and B are N-by-NRHS. and the way she also hands you error bounds and a condition estimate?? multitasking queen 👑 not just solving, but also telling you how much you should trust the answer. obsessed.

🌈 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').
!>
!> 2. 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.
!>
!> 3. 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.
!>
!> 4. The system of equations is solved for X using the factored form
!>    of A.
!>
!> 5. Iterative refinement is applied to improve the computed solution
!>    matrix and calculate error bounds and backward error estimates
!>    for it.
!>
!> 6. 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.
!>

okay so here's the whole glow-up routine, step by step, no skips:

  1. if FACT = 'E', real scaling factors get computed to equilibrate the system — and the formulas are giving math-girlie excellence:
    • 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 the system actually gets equilibrated depends on the scaling of the matrix A (so it's a vibe check, not a promise 💅). but if equilibration IS used, A is overwritten by diag(R)*A*diag(C), and B gets overwritten by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C'). so yeah — heads up, she's modifying your arrays, that's just the deal 🙈

  1. if FACT = 'N' or 'E', the LU decomposition factors 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. classic.
  2. if some U(i,i)=0, making U exactly singular, the routine returns with INFO = i. otherwise, the factored form of A is used to estimate the condition number of A. and here's the tea ☕: if the reciprocal of the condition number is less than machine precision, INFO = N+1 comes back as a warning, but she still keeps going — solving for X and computing error bounds exactly like described below. not her quitting on you, just a little flag 🚩
  3. the system of equations is solved for X using the factored form of A. slay.
  4. iterative refinement is applied to improve the computed solution matrix, and to calculate error bounds and backward error estimates for it. she's refining, she's improving, she's glowing ✨
  5. 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 from before equilibration. bringing it all back home, we love a full circle moment 🌸

🌟 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.
!>

FACT is CHARACTER*1 and she's the main character here — she decides whether you're handing over the already-factored form of the matrix A on entry, and if you're not, whether A should be equilibrated before it gets factored. choices!!

  • = 'F': on entry, AF and IPIV already 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. and good news: A, AF, and IPIV are not modified 🫶
  • = 'N': the matrix A will be copied to AF and factored. simple, clean.
  • = 'E': the matrix A will be equilibrated if necessary, then copied to AF and factored. the full treatment 💅

💙 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)
!>

TRANS is CHARACTER*1 and tells us which form of the system of equations we're on:

  • = 'N': A * X = B (no transpose)
  • = 'T': A**T * X = B (transpose)
  • = 'C': A**H * X = B (transpose)

bestie, if you noticed 'T' and 'C' both say transpose — yeah, that's what the page says, I'm not editorializing, I'm just reporting 👀

💚 N

!>          N is INTEGER
!>          The number of linear equations, i.e., the order of the
!>          matrix A.  N >= 0.
!>

N is INTEGER: the number of linear equations, aka the order of the matrix A. requirement: N >= 0. (zero is allowed, she's flexible 🧚)

💛 NRHS

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

NRHS is INTEGER: the number of right hand sides, aka the number of columns of the matrices B and X. requirement: NRHS >= 0. more right-hand sides = more problems solved per call, efficiency queen 👑

🧡 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).
!>

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 already 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 gets scaled like this:

  • EQUED = 'R': A := diag(R) * A
  • EQUED = 'C': A := A * diag(C)
  • EQUED = 'B': A := diag(R) * A * diag(C).

so yeah, she may rearrange your matrix's whole look. it's a makeover 💅

💗 LDA

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

LDA is INTEGER: the leading dimension of the array A. rule: 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).
!>

AF is DOUBLE PRECISION array, dimension (LDAF,N) and honestly she's got three whole moods depending on FACT:

  • 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 DGETRF. and 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 the form of the equilibrated matrix).

input sometimes, output sometimes, she contains multitudes 🦋

💞 LDAF

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

LDAF is INTEGER: the leading dimension of the array AF. rule: 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.
!>

IPIV is INTEGER array, dimension (N) — the pivot index gossip column 🔥, again with three moods:

  • 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 DGETRF; 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.
!>

EQUED is CHARACTER*1 and she reports the form of equilibration that was done — the receipts, essentially:

  • = '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)

and the direction of info flow: EQUED is an input argument if FACT = 'F'; otherwise, it's an output argument.

ngl, that typo-ish FACT = 'N with no closing quote in the 'N' line? that's straight from the page, bestie. preserving it exactly because we don't edit the source 💀

💝 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.
!>

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 (she's resting, don't bother her). R is an input argument if FACT = 'F'; otherwise, R is an output argument. and ⚠️ 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.
!>

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. and ⚠️ 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.
!>

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

so like… she might replace your B. did you back it up? (the page doesn't say to, but the page does say she overwrites it 💀)

🎀 LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!>

LDB is INTEGER: the leading dimension of the array B. rule: 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'.
!>

X is DOUBLE PRECISION array, dimension (LDX,NRHS) — the star of the show, the actual answer 💖. if INFO = 0 or INFO = N+1, this holds 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'. read that twice, it's load-bearing 🫶

🧚 LDX

!>          LDX is INTEGER
!>          The leading dimension of the array X.  LDX >= max(1,N).
!>

LDX is INTEGER: the leading dimension of the array X. rule: 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.
!>

RCOND is DOUBLE PRECISION: the estimate of the reciprocal condition number of the matrix A after equilibration (if done). and here's the important part — if RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. 😭 that condition gets flagged by a return code of INFO > 0. so RCOND is basically the trust-o-meter. low number = be scared.

🦋 FERR

💖 FERR 💖

FERR is DOUBLE PRECISION array, dimension (NRHS). It's 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. Cautious queen behavior, honestly. 😌

🦋 BERR 🦋

BERR is DOUBLE PRECISION array, dimension (NRHS). This one 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. That's the vibe check for backward error, periodt. ✨

🎀 WORK 🎀

WORK is DOUBLE PRECISION array, dimension (MAX(1,4*N)).

On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). Heads up: the source literally says The norm is used. — keep that exactly in mind, bestie. 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. No cap, that's the tea. 🫖

🌸 IWORK 🌸

IWORK is INTEGER array, dimension (N).

🔥 INFO 🔥

INFO is INTEGER.

  • = 0: successful exit. We love a clean 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.

👑 Author 👑

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

Definition at line 346 of file dgesvx.f.

🤖 Author (the auto-generated one) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

💖 That's the whole page, bestie. This one is LAPACK Version 3.12.0, SRC/dgesvx.f(3) — go off and solve responsibly! 🎀✨

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