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SUBROUTINE <a name="ZSYSVX.1"></a><a href="zsysvx.f.html#ZSYSVX.1">ZSYSVX</a>( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
$ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
$ RWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
DOUBLE PRECISION RCOND
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ WORK( * ), X( LDX, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="ZSYSVX.24"></a><a href="zsysvx.f.html#ZSYSVX.1">ZSYSVX</a> uses the diagonal pivoting factorization to compute the
</span><span class="comment">*</span><span class="comment"> solution to a complex system of linear equations A * X = B,
</span><span class="comment">*</span><span class="comment"> where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
</span><span class="comment">*</span><span class="comment"> matrices.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Error bounds on the solution and a condition estimate are also
</span><span class="comment">*</span><span class="comment"> provided.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Description
</span><span class="comment">*</span><span class="comment"> ===========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The following steps are performed:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
</span><span class="comment">*</span><span class="comment"> The form of the factorization is
</span><span class="comment">*</span><span class="comment"> A = U * D * U**T, if UPLO = 'U', or
</span><span class="comment">*</span><span class="comment"> A = L * D * L**T, if UPLO = 'L',
</span><span class="comment">*</span><span class="comment"> where U (or L) is a product of permutation and unit upper (lower)
</span><span class="comment">*</span><span class="comment"> triangular matrices, and D is symmetric and block diagonal with
</span><span class="comment">*</span><span class="comment"> 1-by-1 and 2-by-2 diagonal blocks.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
</span><span class="comment">*</span><span class="comment"> returns with INFO = i. Otherwise, the factored form of A is used
</span><span class="comment">*</span><span class="comment"> to estimate the condition number of the matrix A. If the
</span><span class="comment">*</span><span class="comment"> reciprocal of the condition number is less than machine precision,
</span><span class="comment">*</span><span class="comment"> INFO = N+1 is returned as a warning, but the routine still goes on
</span><span class="comment">*</span><span class="comment"> to solve for X and compute error bounds as described below.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 3. The system of equations is solved for X using the factored form
</span><span class="comment">*</span><span class="comment"> of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 4. Iterative refinement is applied to improve the computed solution
</span><span class="comment">*</span><span class="comment"> matrix and calculate error bounds and backward error estimates
</span><span class="comment">*</span><span class="comment"> for it.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> FACT (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Specifies whether or not the factored form of A has been
</span><span class="comment">*</span><span class="comment"> supplied on entry.
</span><span class="comment">*</span><span class="comment"> = 'F': On entry, AF and IPIV contain the factored form
</span><span class="comment">*</span><span class="comment"> of A. A, AF and IPIV will not be modified.
</span><span class="comment">*</span><span class="comment"> = 'N': The matrix A will be copied to AF and factored.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> UPLO (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'U': Upper triangle of A is stored;
</span><span class="comment">*</span><span class="comment"> = 'L': Lower triangle of A is stored.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of linear equations, i.e., the order of the
</span><span class="comment">*</span><span class="comment"> matrix A. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NRHS (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of right hand sides, i.e., the number of columns
</span><span class="comment">*</span><span class="comment"> of the matrices B and X. NRHS >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input) COMPLEX*16 array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment"> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
</span><span class="comment">*</span><span class="comment"> upper triangular part of A contains the upper triangular part
</span><span class="comment">*</span><span class="comment"> of the matrix A, and the strictly lower triangular part of A
</span><span class="comment">*</span><span class="comment"> is not referenced. If UPLO = 'L', the leading N-by-N lower
</span><span class="comment">*</span><span class="comment"> triangular part of A contains the lower triangular part of
</span><span class="comment">*</span><span class="comment"> the matrix A, and the strictly upper triangular part of A is
</span><span class="comment">*</span><span class="comment"> not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
</span><span class="comment">*</span><span class="comment"> If FACT = 'F', then AF is an input argument and on entry
</span><span class="comment">*</span><span class="comment"> contains the block diagonal matrix D and the multipliers used
</span><span class="comment">*</span><span class="comment"> to obtain the factor U or L from the factorization
</span><span class="comment">*</span><span class="comment"> A = U*D*U**T or A = L*D*L**T as computed by <a name="ZSYTRF.97"></a><a href="zsytrf.f.html#ZSYTRF.1">ZSYTRF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'N', then AF is an output argument and on exit
</span><span class="comment">*</span><span class="comment"> returns the block diagonal matrix D and the multipliers used
</span><span class="comment">*</span><span class="comment"> to obtain the factor U or L from the factorization
</span><span class="comment">*</span><span class="comment"> A = U*D*U**T or A = L*D*L**T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDAF (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array AF. LDAF >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IPIV (input or output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> If FACT = 'F', then IPIV is an input argument and on entry
</span><span class="comment">*</span><span class="comment"> contains details of the interchanges and the block structure
</span><span class="comment">*</span><span class="comment"> of D, as determined by <a name="ZSYTRF.110"></a><a href="zsytrf.f.html#ZSYTRF.1">ZSYTRF</a>.
</span><span class="comment">*</span><span class="comment"> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
</span><span class="comment">*</span><span class="comment"> interchanged and D(k,k) is a 1-by-1 diagonal block.
</span><span class="comment">*</span><span class="comment"> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
</span><span class="comment">*</span><span class="comment"> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
</span><span class="comment">*</span><span class="comment"> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
</span><span class="comment">*</span><span class="comment"> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
</span><span class="comment">*</span><span class="comment"> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'N', then IPIV is an output argument and on exit
</span><span class="comment">*</span><span class="comment"> contains details of the interchanges and the block structure
</span><span class="comment">*</span><span class="comment"> of D, as determined by <a name="ZSYTRF.121"></a><a href="zsytrf.f.html#ZSYTRF.1">ZSYTRF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input) COMPLEX*16 array, dimension (LDB,NRHS)
</span><span class="comment">*</span><span class="comment"> The N-by-NRHS right hand side matrix B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array B. LDB >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> X (output) COMPLEX*16 array, dimension (LDX,NRHS)
</span><span class="comment">*</span><span class="comment"> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDX (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array X. LDX >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RCOND (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The estimate of the reciprocal condition number of the matrix
</span><span class="comment">*</span><span class="comment"> A. If RCOND is less than the machine precision (in
</span><span class="comment">*</span><span class="comment"> particular, if RCOND = 0), the matrix is singular to working
</span><span class="comment">*</span><span class="comment"> precision. This condition is indicated by a return code of
</span><span class="comment">*</span><span class="comment"> INFO > 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> FERR (output) DOUBLE PRECISION array, dimension (NRHS)
</span><span class="comment">*</span><span class="comment"> The estimated forward error bound for each solution vector
</span><span class="comment">*</span><span class="comment"> X(j) (the j-th column of the solution matrix X).
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