zppsvx (l)  Linux Manuals
zppsvx: 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,
NAME
ZPPSVX  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,SYNOPSIS
 SUBROUTINE ZPPSVX(
 FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
 CHARACTER EQUED, FACT, UPLO
 INTEGER INFO, LDB, LDX, N, NRHS
 DOUBLE PRECISION RCOND
 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
PURPOSE
ZPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equationsA
Error bounds on the solution and a condition estimate are also provided.
DESCRIPTION
The following steps are performed:1. If FACT = aqEaq, real scaling factors are computed to equilibrate
the system:
diag(S)
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)
2. If FACT = aqNaq or aqEaq, the Cholesky decomposition is used to
factor the matrix A
A
A
where U is an upper triangular matrix, L is a lower triangular
matrix, and aq indicates conjugate transpose.
3. If the leading ibyi principal minor is not positive definite,
then the routine returns with INFO
form of A is used to estimate the condition number of the matrix
A.
precision, INFO
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)
equilibration.
ARGUMENTS
 FACT (input) 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.
= aqFaq: On entry, AFP contains the factored form of A.
If EQUED = aqYaq, the matrix A has been equilibrated
with scaling factors given by S. AP and AFP will not
be modified.
= aqNaq: The matrix A will be copied to AFP and factored.
= aqEaq: The matrix A will be equilibrated if necessary, then copied to AFP and factored.  UPLO (input) CHARACTER*1

= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.  N (input) INTEGER
 The number of linear equations, i.e., the order of the matrix A. N >= 0.
 NRHS (input) INTEGER
 The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
 AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
 On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array, except if FACT = aqFaq and EQUED = aqYaq, then A must contain the equilibrated matrix diag(S)*A*diag(S). The jth column of A is stored in the array AP as follows: if UPLO = aqUaq, AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = aqLaq, AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. See below for further details. A is not modified if FACT = aqFaq or aqNaq, or if FACT = aqEaq and EQUED = aqNaq on exit. On exit, if FACT = aqEaq and EQUED = aqYaq, A is overwritten by diag(S)*A*diag(S).
 AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
 If FACT = aqFaq, then AFP 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. aqNaq, then AFP is the factored form of the equilibrated matrix A. If FACT = aqNaq, then AFP 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 = aqEaq, then AFP 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 AP for the form of the equilibrated matrix).
 EQUED (input or output) CHARACTER*1

Specifies the form of equilibration that was done.
= aqNaq: No equilibration (always true if FACT = aqNaq).
= aqYaq: Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = aqFaq; otherwise, it is an output argument.  S (input or output) DOUBLE PRECISION array, dimension (N)
 The scale factors for A; not accessed if EQUED = aqNaq. S is an input argument if FACT = aqFaq; otherwise, S is an output argument. If FACT = aqFaq and EQUED = aqYaq, each element of S must be positive.
 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
 On entry, the NbyNRHS right hand side matrix B. On exit, if EQUED = aqNaq, B is not modified; if EQUED = aqYaq, B is overwritten by diag(S) * B.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 X (output) COMPLEX*16 array, dimension (LDX,NRHS)
 If INFO = 0 or INFO = N+1, the NbyNRHS solution matrix X to the original system of equations. Note that if EQUED = aqYaq, A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
 LDX (input) INTEGER
 The leading dimension of the array X. LDX >= max(1,N).
 RCOND (output) 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 (output) DOUBLE PRECISION array, dimension (NRHS)
 The estimated forward error bound for each solution vector X(j) (the jth 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 (output) 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 (workspace) COMPLEX*16 array, dimension (2*N)
 RWORK (workspace) DOUBLE PRECISION array, dimension (N)
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith 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.
FURTHER DETAILS
The packed storage scheme is illustrated by the following example when N = 4, UPLO = aqUaq:Twodimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]