
CSPSVX(l) ) CSPSVX(l)
NAME
CSPSVX  use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute
the solution to a complex system of linear equations A * X = B, where A is an NbyN sym
metric matrix stored in packed format and X and B are NbyNRHS matrices
SYNOPSIS
SUBROUTINE CSPSVX( 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
CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute
the solution to a complex system of linear equations A * X = B, where A is an NbyN sym
metric matrix stored in packed format and X and B are NbyNRHS 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**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is symmetric and block diagonal with
1by1 and 2by2 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 supplied on entry. =
'F': On entry, AFP and IPIV contain the factored form of A. AP, 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 symmetric matrix A, packed columnwise in a lin
ear array. The jth column of A is stored in the array AP as follows: if UPLO =
'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(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 contains the block diag
onal matrix D and the multipliers used to obtain the factor U or L from the fac
torization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, 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 diag
onal matrix D and the multipliers used to obtain the factor U or L from the fac
torization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed
triangular 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 contains details of the
interchanges and the block structure of D, as determined by CSPTRF. If IPIV(k) >
0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1by1
diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns
k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 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 CSPTRF.
B (input) COMPLEX array, dimension (LDB,NRHS)
The NbyNRHS 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 NbyNRHS 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 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 esti
mate is as reliable as the estimate for RCOND, and is almost always a slight over
estimate of the true error.
BERR (output) 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 solu
tion).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL 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: D(i,i) is exactly zero. The factorization has been completed but the fac
tor 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. Neverthe
less, 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':
Twodimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = 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 CSPSVX(l) 
