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RedHat 9 (Linux i386) - man page for sspsvx (redhat section l)

SSPSVX(l)					)					SSPSVX(l)

NAME
       SSPSVX  -  use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute
       the solution to a real system of linear equations A * X = B, where A is an N-by-N  symmet-
       ric matrix stored in packed format and X and B are N-by-NRHS matrices

SYNOPSIS
       SUBROUTINE SSPSVX( FACT,  UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
			  WORK, IWORK, INFO )

	   CHARACTER	  FACT, UPLO

	   INTEGER	  INFO, LDB, LDX, N, NRHS

	   REAL 	  RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

	   REAL 	  AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ),  WORK(	*  ),  X(
			  LDX, * )

PURPOSE
       SSPSVX  uses  the  diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute
       the solution to a real system of linear equations A * X = B, where A is an N-by-N  symmet-
       ric  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**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
	  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  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) REAL array, dimension (N*(N+1)/2)
	       The upper or lower triangle of the symmetric matrix A, packed columnwise in a lin-
	       ear 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) REAL 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**T or A = L*D*L**T as computed by SSPTRF, 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 SSPTRF, 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 SSPTRF.  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 SSPTRF.

       B       (input) REAL 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) REAL 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 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) REAL array, dimension (3*N)

       IWORK   (workspace) INTEGER 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 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':

       Two-dimensional 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 				SSPSVX(l)


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