cppsvx.f man page

cppsvx.f —

Synopsis

Functions/Subroutines

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

Function/Subroutine Documentation

subroutine cppsvx (characterFACT, characterUPLO, integerN, integerNRHS, complex, dimension( * )AP, complex, dimension( * )AFP, characterEQUED, real, dimension( * )S, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, complex, dimension( * )WORK, real, dimension( * )RWORK, integerINFO)

CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

CPPSVX 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 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 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 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 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 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

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 cppsvx.f.

Author

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Referenced By

cppsvx(3) is an alias of cppsvx.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK