realPOsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

✨ realPOsolve(3) — LAPACK ✨

okay so bestie, welcome to realPOsolve(3) from LAPACK 💖 this page is all about the real solve driver functions for PO matrices, and it's literally giving math girlboss energy. no cap.

📛 NAME 📛

realPOsolve - real 🫶

📜 SYNOPSIS 📜

Functions

  • subroutine sposv (UPLO, N, NRHS, A, LDA, B, LDB, INFO) — SPOSV computes the solution to system of linear equations A * X = B for PO matrices ✨
  • subroutine sposvx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) — SPOSVX computes the solution to system of linear equations A * X = B for PO matrices 💅
  • subroutine sposvxx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) — SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices 👑

🫶 Detailed Description 🫶

This is the group of real solve driver functions for PO matrices. periodt 💅

🔥 Function Documentation 🔥

subroutine sposv (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine sposv (character UPLO, integer N, integer NRHS, real, dimension(
    lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB,
    integer INFO)

SPOSV computes the solution to system of linear equations A * X = B for PO matrices ✨

Purpose

SPOSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as A = U**T* U, if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B. 💖

Parameters

  • UPLO — UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
  • 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 REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.

  • LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.
  • LDB — LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine sposvx (character FACT, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, character EQUED, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

subroutine sposvx (character FACT, character UPLO, integer N, integer NRHS,
    real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF,
    integer LDAF, character EQUED, real, dimension( * ) S, real, dimension(
    ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real
    RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real,
    dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SPOSVX computes the solution to system of linear equations A * X = B for PO matrices 💅

Purpose

SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided.

Description

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system: diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B.
  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix.
  3. If the leading i-by-i principal minor is not positive definite, 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(S) 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 contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be 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.
  • UPLO — UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
  • 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 REAL array, dimension (LDA,N) On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).

  • LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is REAL array, dimension (LDAF,N) If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S).

If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A.

If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T 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).
  • EQUED — EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
  • S — S is REAL array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
  • B — B is REAL 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 EQUED = 'Y', B is overwritten by diag(S) * B.
  • LDB — LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
  • X
  • X — REAL array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X to the original system of equations. Okay so listen: if EQUED = 'Y', then A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. Don't forget that part or you'll be out here debugging the wrong matrix, fr fr 😭
  • LDX — INTEGER. The leading dimension of the array X. Requirement: LDX >= max(1,N). Respect the leading dimension 💅
  • RCOND — REAL. The estimate of the reciprocal condition number of the matrix A after equilibration (if equilibration happened). Here's the tea: if RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. That whole situation gets flagged by a return code of INFO > 0.
  • FERR — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (that's 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). This estimate is as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error — so it's giving cautious, not delulu 🌸
  • BERR — REAL 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. Slay.
  • WORK — REAL array, dimension (3*N). That's the scratch space, bestie, just hand it over.
  • IWORK — INTEGER array, dimension (N). More scratch space, integer edition.
  • INFO — INTEGER. The vibe report 📋:
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 🙈
    • > 0: if INFO = i, and i is
      • <= N: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned.
      • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed, because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. plot twist ✨

✍️ Author ✍️

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

Academic royalty, no cap 👑

🦋 subroutine sposvxx 🦋

subroutine sposvxx (character FACT, character UPLO, integer N, integer NRHS,
    real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF,
    integer LDAF, character EQUED, real, dimension( * ) S, real, dimension(
    ldb, * ) B, integer LDB, real, 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, real,
    dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

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

🌸 Purpose 🌸

SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations

A * X = B,

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

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

SPOSVXX 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 SPOSVXX 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 SPOSVXX would itself produce. So like, if you're rolling your own, that's on you bestie 🫶

🌈 Description 🌈

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*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(S)*A*diag(S) and B by diag(S)*B.

  1. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix.

  1. If the leading i-by-i principal minor is not positive definite, 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. That's the secret sauce, iykyk 💅
  4. If equilibration was used, the matrix X is premultiplied by diag(S) 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. Convenient, right? 🎀

💅 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 contains the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by S. A and AF 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.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • 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 REAL array, dimension (LDA,N). On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S). 👀 so watch out, it comes back different.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is REAL array, dimension (LDAF,N).
    • If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S).
    • If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A.
    • If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T 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).
  • EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done.
    • = 'N': No equilibration (always true if FACT = 'N').
    • = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. Switch-hitter energy 💫

  • S — (REAL array, dimension (N)) 🎀 the row scale factors for A, bestie. If EQUED = 'Y', then A is multiplied on the left and on the right by diag(S) — yeah, both sides, it's giving full glam 💅. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive (no zeroes, no negativity, periodt). If S is output, each element of S is a power of the radix. If S is input, each element of S 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. And 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.
lowkey important: powers of the radix are the vibe here. if you scale by anything else and things underflow/overflow, your error estimates stop being trustworthy. not a cute look. 🙈
  • B — (REAL array, dimension (LDB,NRHS)) 🌸 On entry, this is the N-by-NRHS right hand side matrix B. On exit: if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S)*B.
  • LDB — (INTEGER) the leading dimension of the array B. LDB >= max(1,N). 🦋
  • X — (REAL array, dimension (LDX,NRHS)) ✨ If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Heads up though: A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X.
  • LDX — (INTEGER) the leading dimension of the array X. LDX >= max(1,N). 💜
  • RCOND — (REAL) the 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. But — and this is the plot twist — the error may still be small even if this number is very small and the matrix appears ill-conditioned. 😭
  • RPVGRW — (REAL) reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is the one being used here. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. That also means the solution X, the estimated condition numbers, and the error bounds could all be unreliable. And if factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. 🔥
  • BERR — (REAL 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) the 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 — (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 *i*th 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 *i*th 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) * slamch('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) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — reciprocal condition number: the estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('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 — (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 *i*th 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 *i*th 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) * slamch('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) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — reciprocal condition number: the estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('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. 💫
  • PARAMS — (REAL 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: 1.0
      • = 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. 💀
    • 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 — (REAL array, dimension (4*N)) 🫶
  • IWORK — (INTEGER array, dimension (N)) 🫶
  • INFO
  • = 0 — slay, we did it 🎉 Successful exit, and bestie, the solution to every right-hand side is guaranteed. No notes. 💅
  • < 0 — uh oh 🙈 If INFO = -i, that means the i-th argument had an illegal value. Like, the call itself showed up with a bad outfit.
  • > 0 and <= N — so here's the tea 😭 U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, which means the solution and error bounds could not be computed at all, and RCOND = 0 gets returned. Not a vibe. Handle with care.
  • = N+J — okay this one's the messy one, stay with me 🫶 The solution corresponding to 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, because the routine is dramatic and only has room for one confession.
    • If a small componentwise error is not requested (PARAMS(3) = 0.0), then the Jth right-hand side is the first one with a normwise error bound that isn't guaranteed — meaning the smallest J such that ERR_BNDS_NORM(J,1) = 0.0.
    • By default (PARAMS(3) = 1.0), the Jth right-hand side is the first with either a normwise or 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.

See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). And if you're nosy (respectfully 💖) and want info about all of the right-hand sides, go check ERR_BNDS_NORM or ERR_BNDS_COMP.

👑 Author

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

✨ Author (yes, again 💅)

Generated automatically by Doxygen for LAPACK from the source code. Not me writing fanfic — the machine did that. 🤖

and that's a wrap, bestie 🌈✨

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