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SUBROUTINE <a name="SSPSVX.1"></a><a href="sspsvx.f.html#SSPSVX.1">SSPSVX</a>( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
$ LDX, RCOND, FERR, BERR, WORK, IWORK, 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, LDB, LDX, N, NRHS
REAL RCOND
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IPIV( * ), IWORK( * )
REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
$ FERR( * ), 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="SSPSVX.22"></a><a href="sspsvx.f.html#SSPSVX.1">SSPSVX</a> uses the diagonal pivoting factorization A = U*D*U**T or
</span><span class="comment">*</span><span class="comment"> A = L*D*L**T to compute the solution to a real system of linear
</span><span class="comment">*</span><span class="comment"> equations A * X = B, where A is an N-by-N symmetric matrix stored
</span><span class="comment">*</span><span class="comment"> in packed format and X and B are N-by-NRHS 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 as
</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, AFP and IPIV contain the factored form of
</span><span class="comment">*</span><span class="comment"> A. AP, AFP and IPIV will not be modified.
</span><span class="comment">*</span><span class="comment"> = 'N': The matrix A will be copied to AFP 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"> AP (input) REAL array, dimension (N*(N+1)/2)
</span><span class="comment">*</span><span class="comment"> The upper or lower triangle of the symmetric matrix A, packed
</span><span class="comment">*</span><span class="comment"> columnwise in a linear array. The j-th column of A is stored
</span><span class="comment">*</span><span class="comment"> in the array AP as follows:
</span><span class="comment">*</span><span class="comment"> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
</span><span class="comment">*</span><span class="comment"> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
</span><span class="comment">*</span><span class="comment"> See below for further details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AFP (input or output) REAL array, dimension
</span><span class="comment">*</span><span class="comment"> (N*(N+1)/2)
</span><span class="comment">*</span><span class="comment"> If FACT = 'F', then AFP 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="SSPTRF.91"></a><a href="ssptrf.f.html#SSPTRF.1">SSPTRF</a>, stored as
</span><span class="comment">*</span><span class="comment"> a packed triangular matrix in the same storage format as A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'N', then AFP is an output argument and on exit
</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="SSPTRF.97"></a><a href="ssptrf.f.html#SSPTRF.1">SSPTRF</a>, stored as
</span><span class="comment">*</span><span class="comment"> a packed triangular matrix in the same storage format as A.
</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="SSPTRF.103"></a><a href="ssptrf.f.html#SSPTRF.1">SSPTRF</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="SSPTRF.114"></a><a href="ssptrf.f.html#SSPTRF.1">SSPTRF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input) REAL 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) REAL 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) REAL
</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
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