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---
:name: zposvx
:md5sum: 1f3aee2761dee9a6b8c69a6d6362d83d
:category: :subroutine
:arguments:
- fact:
:type: char
:intent: input
- uplo:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- nrhs:
:type: integer
:intent: input
- a:
:type: doublecomplex
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- af:
:type: doublecomplex
:intent: input/output
:dims:
- ldaf
- n
- ldaf:
:type: integer
:intent: input
- equed:
:type: char
:intent: input/output
- s:
:type: doublereal
:intent: input/output
:dims:
- n
- b:
:type: doublecomplex
:intent: input/output
:dims:
- ldb
- nrhs
- ldb:
:type: integer
:intent: input
- x:
:type: doublecomplex
:intent: output
:dims:
- ldx
- nrhs
- ldx:
:type: integer
:intent: input
- rcond:
:type: doublereal
:intent: output
- ferr:
:type: doublereal
:intent: output
:dims:
- nrhs
- berr:
:type: doublereal
:intent: output
:dims:
- nrhs
- work:
:type: doublecomplex
:intent: workspace
:dims:
- 2*n
- rwork:
:type: doublereal
:intent: workspace
:dims:
- n
- info:
:type: integer
:intent: output
:substitutions:
ldx: MAX(1,n)
:fortran_help: " SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to\n\
* compute the solution to a complex system of linear equations\n\
* A * X = B,\n\
* where A is an N-by-N Hermitian positive definite matrix and X and B\n\
* are N-by-NRHS matrices.\n\
*\n\
* Error bounds on the solution and a condition estimate are also\n\
* provided.\n\
*\n\
* Description\n\
* ===========\n\
*\n\
* The following steps are performed:\n\
*\n\
* 1. If FACT = 'E', real scaling factors are computed to equilibrate\n\
* the system:\n\
* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B\n\
* Whether or not the system will be equilibrated depends on the\n\
* scaling of the matrix A, but if equilibration is used, A is\n\
* overwritten by diag(S)*A*diag(S) and B by diag(S)*B.\n\
*\n\
* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to\n\
* factor the matrix A (after equilibration if FACT = 'E') as\n\
* A = U**H* U, if UPLO = 'U', or\n\
* A = L * L**H, if UPLO = 'L',\n\
* where U is an upper triangular matrix and L is a lower triangular\n\
* matrix.\n\
*\n\
* 3. If the leading i-by-i principal minor is not positive definite,\n\
* then the routine returns with INFO = i. Otherwise, the factored\n\
* form of A is used to estimate the condition number of the matrix\n\
* A. If the reciprocal of the condition number is less than machine\n\
* precision, INFO = N+1 is returned as a warning, but the routine\n\
* still goes on to solve for X and compute error bounds as\n\
* described below.\n\
*\n\
* 4. The system of equations is solved for X using the factored form\n\
* of A.\n\
*\n\
* 5. Iterative refinement is applied to improve the computed solution\n\
* matrix and calculate error bounds and backward error estimates\n\
* for it.\n\
*\n\
* 6. If equilibration was used, the matrix X is premultiplied by\n\
* diag(S) so that it solves the original system before\n\
* equilibration.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* FACT (input) CHARACTER*1\n\
* Specifies whether or not the factored form of the matrix A is\n\
* supplied on entry, and if not, whether the matrix A should be\n\
* equilibrated before it is factored.\n\
* = 'F': On entry, AF contains the factored form of A.\n\
* If EQUED = 'Y', the matrix A has been equilibrated\n\
* with scaling factors given by S. A and AF will not\n\
* be modified.\n\
* = 'N': The matrix A will be copied to AF and factored.\n\
* = 'E': The matrix A will be equilibrated if necessary, then\n\
* copied to AF and factored.\n\
*\n\
* UPLO (input) CHARACTER*1\n\
* = 'U': Upper triangle of A is stored;\n\
* = 'L': Lower triangle of A is stored.\n\
*\n\
* N (input) INTEGER\n\
* The number of linear equations, i.e., the order of the\n\
* matrix A. N >= 0.\n\
*\n\
* NRHS (input) INTEGER\n\
* The number of right hand sides, i.e., the number of columns\n\
* of the matrices B and X. NRHS >= 0.\n\
*\n\
* A (input/output) COMPLEX*16 array, dimension (LDA,N)\n\
* On entry, the Hermitian matrix A, except if FACT = 'F' and\n\
* EQUED = 'Y', then A must contain the equilibrated matrix\n\
* diag(S)*A*diag(S). If UPLO = 'U', the leading\n\
* N-by-N upper triangular part of A contains the upper\n\
* triangular part of the matrix A, and the strictly lower\n\
* triangular part of A is not referenced. If UPLO = 'L', the\n\
* leading N-by-N lower triangular part of A contains the lower\n\
* triangular part of the matrix A, and the strictly upper\n\
* triangular part of A is not referenced. A is not modified if\n\
* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.\n\
*\n\
* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by\n\
* diag(S)*A*diag(S).\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,N).\n\
*\n\
* AF (input or output) COMPLEX*16 array, dimension (LDAF,N)\n\
* If FACT = 'F', then AF is an input argument and on entry\n\
* contains the triangular factor U or L from the Cholesky\n\
* factorization A = U**H*U or A = L*L**H, in the same storage\n\
* format as A. If EQUED .ne. 'N', then AF is the factored form\n\
* of the equilibrated matrix diag(S)*A*diag(S).\n\
*\n\
* If FACT = 'N', then AF is an output argument and on exit\n\
* returns the triangular factor U or L from the Cholesky\n\
* factorization A = U**H*U or A = L*L**H of the original\n\
* matrix A.\n\
*\n\
* If FACT = 'E', then AF is an output argument and on exit\n\
* returns the triangular factor U or L from the Cholesky\n\
* factorization A = U**H*U or A = L*L**H of the equilibrated\n\
* matrix A (see the description of A for the form of the\n\
* equilibrated matrix).\n\
*\n\
* LDAF (input) INTEGER\n\
* The leading dimension of the array AF. LDAF >= max(1,N).\n\
*\n\
* EQUED (input or output) CHARACTER*1\n\
* Specifies the form of equilibration that was done.\n\
* = 'N': No equilibration (always true if FACT = 'N').\n\
* = 'Y': Equilibration was done, i.e., A has been replaced by\n\
* diag(S) * A * diag(S).\n\
* EQUED is an input argument if FACT = 'F'; otherwise, it is an\n\
* output argument.\n\
*\n\
* S (input or output) DOUBLE PRECISION array, dimension (N)\n\
* The scale factors for A; not accessed if EQUED = 'N'. S is\n\
* an input argument if FACT = 'F'; otherwise, S is an output\n\
* argument. If FACT = 'F' and EQUED = 'Y', each element of S\n\
* must be positive.\n\
*\n\
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)\n\
* On entry, the N-by-NRHS righthand side matrix B.\n\
* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',\n\
* B is overwritten by diag(S) * B.\n\
*\n\
* LDB (input) INTEGER\n\
* The leading dimension of the array B. LDB >= max(1,N).\n\
*\n\
* X (output) COMPLEX*16 array, dimension (LDX,NRHS)\n\
* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to\n\
* the original system of equations. Note that if EQUED = 'Y',\n\
* A and B are modified on exit, and the solution to the\n\
* equilibrated system is inv(diag(S))*X.\n\
*\n\
* LDX (input) INTEGER\n\
* The leading dimension of the array X. LDX >= max(1,N).\n\
*\n\
* RCOND (output) DOUBLE PRECISION\n\
* The estimate of the reciprocal condition number of the matrix\n\
* A after equilibration (if done). If RCOND is less than the\n\
* machine precision (in particular, if RCOND = 0), the matrix\n\
* is singular to working precision. This condition is\n\
* indicated by a return code of INFO > 0.\n\
*\n\
* FERR (output) DOUBLE PRECISION array, dimension (NRHS)\n\
* The estimated forward error bound for each solution vector\n\
* X(j) (the j-th column of the solution matrix X).\n\
* If XTRUE is the true solution corresponding to X(j), FERR(j)\n\
* is an estimated upper bound for the magnitude of the largest\n\
* element in (X(j) - XTRUE) divided by the magnitude of the\n\
* largest element in X(j). The estimate is as reliable as\n\
* the estimate for RCOND, and is almost always a slight\n\
* overestimate of the true error.\n\
*\n\
* BERR (output) DOUBLE PRECISION array, dimension (NRHS)\n\
* The componentwise relative backward error of each solution\n\
* vector X(j) (i.e., the smallest relative change in\n\
* any element of A or B that makes X(j) an exact solution).\n\
*\n\
* WORK (workspace) COMPLEX*16 array, dimension (2*N)\n\
*\n\
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
* > 0: if INFO = i, and i is\n\
* <= N: the leading minor of order i of A is\n\
* not positive definite, so the factorization\n\
* could not be completed, and the solution has not\n\
* been computed. RCOND = 0 is returned.\n\
* = N+1: U is nonsingular, but RCOND is less than machine\n\
* precision, meaning that the matrix is singular\n\
* to working precision. Nevertheless, the\n\
* solution and error bounds are computed because\n\
* there are a number of situations where the\n\
* computed solution can be more accurate than the\n\
* value of RCOND would suggest.\n\
*\n\n\
* =====================================================================\n\
*\n"
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