CHPSVX(l) ) CHPSVX(l)NAME
CHPSVX - use 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 equa‐
tions 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
SYNOPSIS
SUBROUTINE CHPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * )
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, *
)
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 equa‐
tions 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 solu‐
tion 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.
ARGUMENTS
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been sup‐
plied 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 (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': 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) 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 (input or output) COMPLEX array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on entry con‐
tains 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 triangu‐
lar matrix in the same storage format as A.
If FACT = 'N', then AFP is an output argument and on exit con‐
tains 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 triangu‐
lar matrix in the same storage format as A.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry con‐
tains 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 con‐
tains details of the interchanges and the block structure of D,
as determined by CHPTRF.
B (input) COMPLEX array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) 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 (output) 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 (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vec‐
tor 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 array, dimension (2*N)
RWORK (workspace) REAL array, dimension (N)
INFO (output) 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 com‐
pleted 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 preci‐
sion, 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 = '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 ]
LAPACK version 3.0 15 June 2000 CHPSVX(l)