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chpsvx-f(3)

chpsvx.f –

Synopsis

Functions/Subroutines

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

Function/Subroutine Documentation

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

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

Purpose:

 CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
 A = L*D*L**H to compute the solution to a complex system of linear
 equations A * X = B, where A is an N-by-N Hermitian 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 = 'N', the diagonal pivoting method is used to factor A as
 A = U * D * U**H, if UPLO = 'U', or
 A = L * D * L**H, if UPLO = 'L',
 where U (or L) is a product of permutation and unit upper (lower)
 triangular matrices and D is Hermitian and block diagonal with
 1-by-1 and 2-by-2 diagonal blocks.
 2. If some D(i,i)=0, so that D is exactly singular, 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.
 3. The system of equations is solved for X using the factored form
 of A.
 4. Iterative refinement is applied to improve the computed solution
 matrix and calculate error bounds and backward error estimates
 for it.

 

Parameters:

FACT

 FACT is CHARACTER*1
 Specifies whether or not the factored form of A has been
 supplied on entry.
 = 'F': On entry, AFP and IPIV contain the factored form of
 A. AFP and IPIV will not be modified.
 = 'N': The matrix A will be 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)
 The upper or lower triangle of the Hermitian matrix A, packed
 columnwise in a linear array. 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
 See below for further details.

AFP

 AFP is COMPLEX array, dimension (N*(N+1)/2)
 If FACT = 'F', then AFP is an input argument and on entry
 contains the block diagonal matrix D and the multipliers used
 to obtain the factor U or L from the factorization
 A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as
 a packed triangular matrix in the same storage format as A.
 If FACT = 'N', then AFP is an output argument and on exit
 contains the block diagonal matrix D and the multipliers used
 to obtain the factor U or L from the factorization
 A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as
 a packed triangular matrix in the same storage format as A.

IPIV

 IPIV is INTEGER array, dimension (N)
 If FACT = 'F', then IPIV is an input argument and on entry
 contains details of the interchanges and the block structure
 of D, as determined by CHPTRF.
 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
 interchanged and D(k,k) is a 1-by-1 diagonal block.
 If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
 columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
 is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
 IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
 interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
 If FACT = 'N', then IPIV is an output argument and on exit
 contains details of the interchanges and the block structure
 of D, as determined by CHPTRF.

B

 B is COMPLEX 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 array, dimension (LDX,NRHS)
 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix 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. 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: D(i,i) is exactly zero. The factorization
 has been completed but the factor D is exactly
 singular, so the solution and error bounds could
 not be computed. RCOND = 0 is returned.
 = N+1: D 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 277 of file chpsvx.f.

Author

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