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zppsvx.f(3) LAPACK zppsvx.f(3)

zppsvx.f -


subroutine zppsvx (FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
 
ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

ZPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
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 equations
    A * X = B,
 where A is an N-by-N Hermitian positive definite matrix stored in
 packed format 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**H * U , if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix, L is a lower triangular matrix, and **H indicates conjugate transpose.
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, AFP contains the factored form of A.
                  If EQUED = 'Y', the matrix A has been equilibrated
                  with scaling factors given by S.  AP and AFP will not
                  be modified.
          = 'N':  The matrix A will be copied to AFP and factored.
          = 'E':  The matrix A will be equilibrated if necessary, then
                  copied to AFP 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.
AP
          AP is 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 = 'F'
          and EQUED = 'Y', then A must contain the equilibrated matrix
          diag(S)*A*diag(S).  The j-th column of A is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.  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).
AFP
          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
          If FACT = 'F', 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. 'N', then AFP is the factored
          form of the equilibrated matrix A.
If FACT = 'N', 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 = 'E', 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
          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 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 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.
Date:
April 2012
Further Details:
  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14 a22 a23 a24 a33 a34 (aij = conjg(aji)) a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
Definition at line 311 of file zppsvx.f.

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Sat Nov 16 2013 Version 3.4.2

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