complex16POsolve(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 complex16POsolve(3) 💖

okay so bestie, welcome to complex16POsolve from the LAPACK manual — this is the complex16 PO-matrix solve driver crew and they are so iconic, no cap. it's the whole squad of routines that solve A * X = B when A is Hermitian positive definite and complex. buckle up, we're doing four parts and this is part one 🎀

✨ NAME ✨

complex16POsolve — complex16

🦋 SYNOPSIS 🦋

🌸 Functions 🌸

subroutine zcposv (UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK,
    RWORK, ITER, INFO)

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

subroutine zposv (UPLO, N, NRHS, A, LDA, B, LDB, INFO)

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

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

subroutine zposvxx (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, RWORK, INFO)

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

🌈 Detailed Description 🌈

This is the group of complex16 solve driver functions for PO matrices. the whole girlband, together 💅

🔥 Function Documentation 🔥

👑 subroutine `zcposv` 👑

subroutine zcposv (character UPLO, integer N, integer NRHS, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
    integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, complex*16,
    dimension( n, * ) WORK, complex, dimension( * ) SWORK, double precision,
    dimension( * ) RWORK, integer ITER, integer INFO)

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

💫 Purpose 💫

ZCPOSV computes 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.

ZCPOSV first attempts to factorize the matrix in COMPLEX and use this factorization within an iterative refinement procedure to produce a solution with COMPLEX*16 normwise backward error quality (see below). If the approach fails the method switches to a COMPLEX*16 factorization and solve.

The iterative refinement is not going to be a winning strategy if the ratio COMPLEX performance over COMPLEX*16 performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement. (iconic behaviour honestly, always shooting her shot ✨)

The iterative refinement process is stopped if

    ITER > ITERMAX

or for all the RHS we have:

    RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where

ITER is the number of the current iteration in the iterative refinement process
RNRM is the infinity-norm of the residual
XNRM is the infinity-norm of the solution
ANRM is the infinity-operator-norm of the matrix A
EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.

🌟 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 COMPLEX*16 array, dimension (LDA,N).

On entry, the Hermitian 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.

Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero.

On exit, if iterative refinement has been successfully used (INFO = 0 and ITER >= 0, see description below), then A is unchanged, if double precision factorization has been used (INFO = 0 and ITER < 0, see description below), then the array A contains the factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H.

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX*16 array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, the N-by-NRHS solution matrix X.
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
WORK — WORK is COMPLEX*16 array, dimension (N,NRHS). This array is used to hold the residual vectors.
SWORK — SWORK is COMPLEX array, dimension (N*(N+NRHS)). This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
RWORK — RWORK is DOUBLE PRECISION array, dimension (N).
ITER — ITER is INTEGER
- < 0: iterative refinement has failed, COMPLEX*16 factorization has been performed
  - -1 : the routine fell back to full precision for implementation- or machine-specific reasons
  - -2 : narrowing the precision induced an overflow, the routine fell back to full precision
  - -3 : failure of CPOTRF
  - -31: stop the iterative refinement after the 30th iterations
- > 0: iterative refinement has been successfully used. Returns the number of iterations
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 (COMPLEX*16) 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 `zposv` 👑

subroutine zposv (character UPLO, integer N, integer NRHS, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
    integer LDB, integer INFO)

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

💫 Purpose 💫

ZPOSV computes 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.

The Cholesky decomposition is used to factor A 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. 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 COMPLEX*16 array, dimension (LDA,N).

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

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX*16 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 `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

💫 Purpose 💫

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.

bestie we are so not done — zposvx still has her whole parameter list, and there's a whole zposvxx waiting in part 2 💖 don't go anywhere ✨

Okay so error bounds on the solution and a condition estimate are also provided, bestie. 💖 Like, the whole package.

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 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.
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 is INTEGER.
- = 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.

Author 🌸

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

subroutine zposvxx 🎀

   subroutine zposvxx (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 RPVGRW, double precision, dimension( * ) BERR, integer
       N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double
       precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double
       precision, dimension( * ) PARAMS, complex*16, dimension( * ) WORK,
       double precision, dimension( * ) RWORK, integer INFO)

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

Purpose ✨

ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a complex*16 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.

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

ZPOSVXX is also totally fine with you rolling up with your own factorization and equilibration factors — just peep the deets on the FACT and EQUED options before you do. And bestie, if you solve with refinement using a factorization from an earlier ZPOSVXX call, you'll also get a solution with either O(eps) errors or warnings. But we cannot — cannot — make that promise for random general user-provided factorizations and equilibration factors that differ from what ZPOSVXX would've cooked up itself. 🙈

✨ Description ✨

Here's the whole glow-up sequence, step by step, no skips:

If FACT = 'E', double precision scaling factors get computed to equilibrate the system:

  diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B

Whether or not the system actually gets 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 factors 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 i-by-i principal minor is not positive definite, the routine bounces with INFO = i. Otherwise, the factored form of A gets used to estimate the condition number of the matrix A (see the argument RCOND). And 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 like described below. Iconic stubbornness, honestly.
The system of equations gets solved for X using the factored form of A.
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. ✨
If equilibration was used, the matrix X gets 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. And if the defaults are acceptable, you can pass NPARAMS = 0, which stops the source code from even touching the PARAMS argument. Efficiency queen behavior. 👑

🎀 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 COMPLEX*16 array, dimension (LDA,N)

On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED = 'Y', in which case 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 it 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 it 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 it 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's an output argument.

`S`

S is DOUBLE PRECISION array, dimension (N)

The row scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). 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. 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. 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. 😭

`B`

B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)

If INFO = 0, the N-by-NRHS solution matrix X to the original system of equations. Heads up: A and B are modified on exit if EQUED .ne. 'N', 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

Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if it happened). If this is less than the machine precision (in particular, if it's zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. She's complicated like that. 💜

`RPVGRW`

RPVGRW is DOUBLE PRECISION

Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could 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`

BERR is DOUBLE PRECISION array, dimension (NRHS)

Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).

`N_ERR_BNDS`

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 DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains info about various error bounds and condition numbers corresponding to the normwise relative error, which is defined like this:

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) * dlamch('Epsilon').
err = 2 — 'Guaranteed' error bound: The estimated forward error, almost certainly within a factor of 10 of the true error, so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 👑
err = 3 — Reciprocal condition number: the estimated normwise reciprocal condition number. It gets vibe-checked against the threshold sqrt(n) * dlamch('Epsilon') to decide 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. And like, Z = S*A, where S scales each row by a power of the radix so that all the absolute row sums of Z are approximately 1. math girlbossery ✨

See Lapack Working Note 165 for further details and extra cautions. read it bestie, it's giving important 📖

🎀 ERR_BNDS_COMP 🎀

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS).

For each right-hand side, this array is holding info about all the various error bounds and condition numbers that go with the componentwise relative error, which is defined like this:

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 (the componentwise relative error depends on it, bestie) and by the type of error information as described below. There are currently up to three pieces of information returned for each right-hand side. If componentwise accuracy isn't requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. And if N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries get returned. know your limits 💅

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). trust issues but make it numerical 🙈
err = 2 — 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error — so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. no cap, only trust it when the boolean says so
err = 3 — Reciprocal condition number: the estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. iconic scaling behaviour honestly ⭐

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. defaults are the moment ✨

🌸 PARAMS 🌸

PARAMS is DOUBLE PRECISION array, dimension NPARAMS.

Specifies algorithm parameters. If an entry is < 0.0, then that entry gets filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. so if you're lazy, the defaults have your back 🫶

PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not.
- Default: 1.0D+0
- = 0.0: No refinement is performed, and no error bounds are computed.
- = 1.0: Use the extra-precise refinement algorithm.
- (other values are reserved for future use) don't try it 🚫
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. bestie be so serious rn, the guarantees can stop being real 😭
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) slay, we love convergence 💖

🦋 WORK 🦋

WORK is COMPLEX*16 array, dimension (2*N).

🌈 RWORK 🌈

RWORK is DOUBLE PRECISION array, dimension (2*N).

👑 INFO 👑

INFO is INTEGER.

= 0: Successful exit. The solution to every right-hand side is guaranteed. we ate and left no crumbs 🎉
< 0: If INFO = -i, the i-th argument had an illegal value. whoops, illegal 💀
> 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. singular matrix arc, not the good kind 😭
= N+J: The solution corresponding to the Jth 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 Jth 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 Jth 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. only the first problem child gets named, the rest are on you to investigate 🔍

💜 Author 💜

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

Author — Generated automatically by Doxygen for LAPACK from the source code. yes the docs wrote themselves, we love a self-sufficient king ✨

okay that's the whole page bestie!! 🎀 this was LAPACK Version 3.11.0, dated Sun Nov 27 2022, and you were reading the manual page for complex16POsolve(3) — go forth and solve those complex double-precision systems, you absolute legend 👑💖🦋

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complex16POsolve(3)

💖 complex16POsolve(3) 💖

✨ NAME ✨

🦋 SYNOPSIS 🦋

🌸 Functions 🌸

🌈 Detailed Description 🌈

🔥 Function Documentation 🔥

👑 subroutine zcposv 👑

💫 Purpose 💫

🌟 Parameters 🌟

🫶 Author 🫶

👑 subroutine zposv 👑

💫 Purpose 💫

🌟 Parameters 🌟

🫶 Author 🫶

👑 subroutine zposvx 👑

💫 Purpose 💫

Description ✨

Parameters 💅

Author 🌸

subroutine zposvxx 🎀

Purpose ✨

✨ Description ✨

🎀 Parameters 🎀

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 🦋

🌈 RWORK 🌈

👑 INFO 👑

💜 Author 💜

👑 subroutine `zcposv` 👑

👑 subroutine `zposv` 👑

👑 subroutine `zposvx` 👑

`FACT`

`UPLO`

`N`

`NRHS`

`A`

`LDA`

`AF`

`LDAF`

`EQUED`

`S`

`B`

`LDB`

`X`

`LDX`

`RCOND`

`RPVGRW`

`BERR`

`N_ERR_BNDS`

`ERR_BNDS_NORM`