posvx(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

✨ posvx(3) ✨

bestie, this is the Library Functions Manual entry for posvx, and it's literally giving factor and solve, expert energy. no cap, we're in expert mode. 💅

🌸 NAME 🌸

posvx — posvx: factor and solve, expert. periodt.

🎀 SYNOPSIS 🎀

✨ Functions ✨

💅 `cposvx`

       subroutine cposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b,
           ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

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

💅 `dposvx`

       subroutine dposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b,
           ldb, x, ldx, rcond, ferr, berr, work, iwork, info)

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

💅 `sposvx`

       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.

💅 `zposvx`

       subroutine zposvx (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b,
           ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

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

🌈 Detailed Description 🌈

okay so this is the detailed description zone — the actual function documentation is right below, bestie 💖

🧚 Function Documentation 🧚

🌟 subroutine cposvx 🌟

   subroutine cposvx (character fact, character uplo, integer n, integer nrhs,
       complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, *
       ) af, integer ldaf, character equed, real, dimension( * ) s, complex,
       dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x,
       integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * )
       berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer
       info)

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

💖 Purpose 💖

okay so CPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations

A * X = B

where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. Error bounds on the solution and a condition estimate are also provided. no cap, it's giving expert mode. 💅

🎀 Description 🎀

The following steps are performed:

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.

If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = U**H* U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',

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

If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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 COMPLEX array, dimension (LDA,N). On entry, the Hermitian 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 COMPLEX 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**H*U or A = L*L**H, 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**H*U or A = L*L**H 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**H*U or A = L*L**H 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 COMPLEX array, dimension (LDB,NRHS). On entry, the N-by-NRHS righthand 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 is COMPLEX 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 if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. 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 REAL 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 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).
WORK — WORK is COMPLEX array, dimension (2*N).
RWORK — RWORK is REAL array, dimension (N).
INFO — the full description is in the next part, bestie 💅 (hold that thought!)
INFO — an INTEGER, and yes she's the one who tells you how it went 💅
- = 0: iconic, it worked, successful exit ✨
- < 0: messy — if INFO = -i, then the i-th argument had an illegal value (check your inputs bestie 😭)
- > 0: if INFO = i, and i is…
  - <= N: the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. Not the vibe. 💀
  - = 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. So like… she's still gonna try 🫶

🎀 Author

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

📍 Definition

Definition at line 303 of file cposvx.f.

🌸 subroutine dposvx

Okay bestie, next up: the double precision sister 💖 same energy, bigger floats.

subroutine dposvx (character fact, character uplo, integer n, integer nrhs,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldaf, * ) af, integer ldaf, character equed, double
    precision, dimension( * ) s, 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)

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

🎯 Purpose

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

Like, she doesn't just hand you X and dip — she also gives you the receipts on how wrong it might be 💅✨

🦋 Description

Here's the whole glow-up routine, step by step — the following steps are performed:

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. 💫

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. Splitting her into triangles, as one does 🎀

If the leading principal minor of order i is not positive, 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. She's dramatic but she doesn't give up 😤
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Glow-up round two, no cap 🔥
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration. Undoing the makeover so you get your answer back 🫶

💅 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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. She's giving both ways, iykyk 🧚

S — S is DOUBLE PRECISION 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 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 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 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 if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. The main character of the whole page tbh 👑
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. Overestimating a lil bit, honestly relatable 💀
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 (3*N). Scratch space, she's just here to do the work 💪
IWORK — IWORK is INTEGER array, dimension (N). More scratch, but integers this time 🍓
INFO — hold that thought bestie — INFO has a whole breakdown of her own and it's coming right up 💅
INFO — it’s an INTEGER, and here’s the whole mood ring of exit codes:
- = 0: successful exit. slay.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, and i is:
  - <= N: the leading principal minor of order i of A is not positive, 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.

💖 Author

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

Definition at line 304 of file dposvx.f.

🦋 subroutine sposvx

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. iconic behavior, honestly.

✨ 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. bestie, it’s giving “solve it and also tell you how sketchy the solve is.” 💅

🎀 Description

The following steps are performed:

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.

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.

If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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 is REAL 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 if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. 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 REAL 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 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).
WORK — WORK is REAL array, dimension (3*N).
IWORK — IWORK is INTEGER array, dimension (N).
INFO —
INFO is INTEGER — bestie, here's the exit-code tea:
- = 0: successful exit. Slay.
- < 0: if INFO = -i, the i-th argument had an illegal value. Not iconic.
- > 0: if INFO = i, and i is:
  - <= N: the leading principal minor of order i of A is not positive, 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.

💖 Author

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

Definition at line 304 of file sposvx.f.

🦋 subroutine zposvx

   subroutine zposvx (character fact, character uplo, integer n, integer nrhs,
       complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
       ldaf, * ) af, integer ldaf, character equed, double precision,
       dimension( * ) s, complex*16, dimension( ldb, * ) b, integer ldb,
       complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond,
       double precision, dimension( * ) ferr, double precision, dimension( * )
       berr, complex*16, dimension( * ) work, double precision, dimension( * )
       rwork, integer info)

ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices — periodt.

🎯 Purpose

Okay so ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations

   A * X = B,

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

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

✨ Description

The following steps are performed:

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.

If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

   A = U**H* U,  if UPLO = 'U', or
   A = L * L**H,  if UPLO = 'L',

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

If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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 COMPLEX*16 array, dimension (LDA,N). On entry, the Hermitian 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 COMPLEX*16 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**H *U or A = L*L**H, 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**H *U or A = L*L**H 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**H *U or A = L*L**H 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 DOUBLE PRECISION 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 COMPLEX*16 array, dimension (LDB,NRHS). On entry, the N-by-NRHS righthand 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 is COMPLEX*16 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 if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
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 COMPLEX*16 array, dimension (2*N).
RWORK — RWORK is DOUBLE PRECISION array, dimension (N).
INFO —

`INFO` — the receipts, bestie 💅

INFO is INTEGER, and it's the one telling you how the whole thing went. Vibe check:

INFO = 0 — successful exit. Slay, it worked. ✨
INFO < 0 — if INFO = -i, the i-th argument had an illegal value. Not iconic, but at least it tells you which arg flopped. 🙈
INFO > 0 — if INFO = i, and i is:
- <= N — the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. That's a hard no, no cap. 💀
- = 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. So like… it's giving “messy but let her cook.” 😭🔥

Author 🫶

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

Definition 📌

Definition at line 303 of file zposvx.f.

Author, but the Doxygen kind 🤖

Author: Generated automatically by Doxygen for LAPACK from the source code.

That's the whole page, bestie! This posvx(3) page is from LAPACK Version 3.12.0 and was generated automatically by Doxygen from the source code — the footer we've got doesn't print a date, so I'm not gonna invent one. Bye! 💖🎀

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

posvx(3)

✨ posvx(3) ✨

🌸 NAME 🌸

🎀 SYNOPSIS 🎀

✨ Functions ✨

💅 cposvx

💅 dposvx

💅 sposvx

💅 zposvx

🌈 Detailed Description 🌈

🧚 Function Documentation 🧚

🌟 subroutine cposvx 🌟

💖 Purpose 💖

🎀 Description 🎀

🦋 Parameters 🦋

🎀 Author

📍 Definition

🌸 subroutine dposvx

🎯 Purpose

🦋 Description

💅 Parameters

💖 Author

🦋 subroutine sposvx

✨ Purpose

🎀 Description

💅 Parameters

💖 Author

🦋 subroutine zposvx

🎯 Purpose

✨ Description

📋 Parameters

INFO — the receipts, bestie 💅

Author 🫶

Definition 📌

Author, but the Doxygen kind 🤖

💅 `cposvx`

💅 `dposvx`

💅 `sposvx`

💅 `zposvx`

`INFO` — the receipts, bestie 💅