/home/abui.../cgesvxx.f(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ NAME ✨

okay so this page is cgesvxx.f(3) — from the Library Functions Manual — and the NAME entry is literally this file path, bestie:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cgesvxx.f

💅 SYNOPSIS 💅

Functions/Subroutines

Here’s the subroutine lineup, slay:

subroutine CGESVXX (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)

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

🌸 Function/Subroutine Documentation 🌸

Full signature, exact and iconic:

subroutine CGESVXX (character fact, character trans, integer n, integer
    nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension(
    ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character
    equed, real, dimension( * ) r, real, dimension( * ) c, complex,
    dimension( ldb, * ) b, integer ldb, complex, dimension( ldx , * ) x,
    integer ldx, real rcond, real rpvgrw, real, dimension( * ) berr, integer
    n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension(
    nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params,
    complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

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

🔥 Purpose 🔥

okay so CGESVXX 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. 💖

If requested, both normwise and maximum componentwise error bounds are returned. CGESVXX 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. ✨

CGESVXX 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 CGESVXX 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 CGESVXX would itself produce. 🦋

💖 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 (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.
  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.
  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. 🎀

🧚 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)

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 COMPLEX 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 COMPLEX 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 CGETRF. 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 CGETRF; 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 REAL 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

  • C is REAL array, dimension (N) — the column scale factors for A, bestie.
  • If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed at all (she's off duty 💅).
  • 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. No exceptions, sorry.
  • 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. 😭 So like, keep it to powers of the radix, okay?

💜 B

  • B is COMPLEX 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.
  • Constraint, and it's giving non-negotiable: LDB >= max(1,N).

🎀 X

  • X is COMPLEX array, dimension (LDX,NRHS)
  • If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Your answer, your moment, main character energy 👑
  • Heads up though 🫶: 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.
  • Same energy as above: LDX >= max(1,N).

🌸 RCOND

  • RCOND is REAL
  • Reciprocal scaled condition number. It's 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 💀
  • But like — plot twist — 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-scroll just because the number looks tiny 🙈

⭐ RPVGRW

  • RPVGRW is REAL
  • Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. (yep, the page literally leaves a lil gap right there before "norm" — that's just how it is, no cap 😭)
  • If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. Not iconic. Not slay.
  • 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 CGESVX, this quantity is returned in WORK(1).

🧚 BERR

  • BERR is REAL array, dimension (NRHS)
  • Componentwise relative backward error — for each solution vector X(j), this is its componentwise relative backward error (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). Literally the tiniest nudge that makes it perfect 💅

🌈 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 REAL 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 boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
    • err = 2 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) * slamch('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) * slamch('Epsilon') to determine if the error estimate is . (again with the missing word — the page really just drops it there 🙈) 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

  • ERR_BNDS_COMP is REAL 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 boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
    • err = 2 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) * slamch('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) * slamch('Epsilon') to determine if the error estimate is . (same gap, same drama 💀) 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. She's not even looked at. Respect her privacy.

👑 PARAMS

  • PARAMS is REAL array, dimension NPARAMS
  • Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with 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: 1.0
      • = 0.0: No refinement is performed, and no error bounds are computed.
      • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
      • (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. ⚠️ Like, you can go feral, but she's not responsible for what happens.
    • 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

(and this is exactly where this chunk of the page runs out — WORK's whole description is right up next, bestie, hold on 💖)

  • WORK — a COMPLEX array with dimension (2*N) ✨ keep that exactly as written, bestie. No shrinking, no vibes-based resizing.
  • RWORK — a REAL array with dimension (2*N) 🫶 same energy, real numbers only.
  • INFO — an INTEGER, aka the ✨vibe check✨ for the whole call. Here’s the full tea:
    • = 0 — successful exit 🎉 The solution to every right-hand side is guaranteed. Slay.
    • < 0 — if INFO = -i, then the i-th argument had an illegal value 🙈 so like, check your inputs.
    • > 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. Not iconic, but extremely important.
    • = N+J — the solution corresponding to the J-th 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 J-th 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 J-th 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.

Definition at line 538 of file cgesvxx.f.

🪄 Author

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