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DPOSVX(l)			       )			     DPOSVX(l)

NAME
       DPOSVX  -  use  the  Cholesky factorization A = U**T*U or A = L*L**T to
       compute the solution to a real system of linear equations A * X = B,

SYNOPSIS
       SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S,  B,
			  LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )

	   CHARACTER	  EQUED, FACT, UPLO

	   INTEGER	  INFO, LDA, LDAF, LDB, LDX, N, NRHS

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION  A(	 LDA, * ), AF( LDAF, * ), B( LDB, * ),
			  BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX, * )

PURPOSE
       DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to com‐
       pute the solution to a real system of linear equations A * X = B, where
       A is an N-by-N symmetric positive definite matrix and X and B are N-by-
       NRHS matrices.

       Error  bounds  on  the  solution and a condition estimate are also pro‐
       vided.

DESCRIPTION
       The following steps are performed:

       1. If FACT = 'E', real scaling factors are computed to equilibrate
	  the system:
	     diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
	  Whether or not the system will be equilibrated depends on the
	  scaling of the matrix A, but if equilibration is used, A is
	  overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

       2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
	  factor the matrix A (after equilibration if FACT = 'E') as
	     A = U**T* U,  if UPLO = 'U', or
	     A = L * L**T,  if UPLO = 'L',
	  where U is an upper triangular matrix and L is a lower triangular
	  matrix.

       3. If the leading i-by-i principal minor is not positive definite,
	  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.

       4. The system of equations is solved for X using the factored form
	  of A.

       5. Iterative refinement is applied to improve the computed solution
	  matrix and calculate error bounds and backward error estimates
	  for it.

       6. If equilibration was used, the matrix X is premultiplied by
	  diag(S) so that it solves the original system before
	  equilibration.

ARGUMENTS
       FACT    (input) CHARACTER*1
	       Specifies whether or not the factored form of the matrix	 A  is
	       supplied	 on  entry, and if not, whether the matrix A should be
	       equilibrated before it is factored.  = 'F':  On entry, AF  con‐
	       tains the factored form of A.  If EQUED = 'Y', the matrix A has
	       been equilibrated with scaling factors given by S.   A  and  AF
	       will  not  be modified.	= 'N':	The matrix A will be copied to
	       AF and factored.
	       = 'E':  The matrix A will be equilibrated  if  necessary,  then
	       copied to AF 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.

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
	       On  entry,  the	symmetric  matrix  A, except if FACT = 'F' and
	       EQUED = 'Y',  then  A  must  contain  the  equilibrated	matrix
	       diag(S)*A*diag(S).   If	UPLO  =	 'U', the leading N-by-N upper
	       triangular part of A contains the upper triangular part of  the
	       matrix  A,  and	the strictly lower triangular part of A is not
	       referenced.  If UPLO = 'L', the leading N-by-N lower triangular
	       part  of	 A contains the lower triangular part of the matrix A,
	       and the strictly upper triangular part of A is not  referenced.
	       A  is  not  modified if FACT = 'F' or 'N', or if FACT = 'E' and
	       EQUED = 'N' on exit.

	       On exit, if FACT = 'E' and EQUED = 'Y',	A  is  overwritten  by
	       diag(S)*A*diag(S).

       LDA     (input) INTEGER
	       The leading dimension of the array A.  LDA >= max(1,N).

       AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
	       If  FACT	 = 'F', then AF is an input argument and on entry con‐
	       tains the triangular factor U or L from the Cholesky factoriza‐
	       tion A = U**T*U or A = L*L**T, in the same storage format as A.
	       If EQUED .ne. 'N', then AF is the factored form of the  equili‐
	       brated matrix diag(S)*A*diag(S).

	       If  FACT	 =  'N',  then	AF  is	an output argument and on exit
	       returns the triangular factor U or L from the Cholesky  factor‐
	       ization A = U**T*U or A = L*L**T of the original matrix A.

	       If  FACT	 =  'E',  then	AF  is	an output argument and on exit
	       returns the triangular factor U or L from the Cholesky  factor‐
	       ization	A  = U**T*U or A = L*L**T of the equilibrated matrix A
	       (see the description of A for  the  form	 of  the  equilibrated
	       matrix).

       LDAF    (input) INTEGER
	       The leading dimension of the array AF.  LDAF >= max(1,N).

       EQUED   (input or output) CHARACTER*1
	       Specifies  the form of equilibration that was done.  = 'N':  No
	       equilibration (always true if FACT = 'N').
	       = 'Y':  Equilibration was done, i.e., A has  been  replaced  by
	       diag(S)	*  A  * diag(S).  EQUED is an input argument if FACT =
	       'F'; otherwise, it is an output argument.

       S       (input or output) DOUBLE PRECISION array, dimension (N)
	       The scale factors for A; not accessed if EQUED = 'N'.  S is  an
	       input  argument	if FACT = 'F'; otherwise, S is an output argu‐
	       ment.  If FACT = 'F' and EQUED = 'Y', each element of S must be
	       positive.

       B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
	       On  entry, the N-by-NRHS right hand side matrix B.  On exit, if
	       EQUED = 'N', B is not modified; if EQUED = 'Y', B is  overwrit‐
	       ten by diag(S) * B.

       LDB     (input) INTEGER
	       The leading dimension of the array B.  LDB >= max(1,N).

       X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
	       If  INFO	 = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
	       the original system of equations.  Note that if EQUED = 'Y',  A
	       and  B  are  modified  on exit, and the solution to the equili‐
	       brated system is inv(diag(S))*X.

       LDX     (input) INTEGER
	       The leading dimension of the array X.  LDX >= max(1,N).

       RCOND   (output) DOUBLE PRECISION
	       The estimate of the reciprocal condition number of the matrix A
	       after  equilibration  (if  done).   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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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:  the leading minor of order i of A is not positive	 defi‐
	       nite,  so  the  factorization  could  not be completed, and the
	       solution has not been computed. RCOND = 0 is returned.  =  N+1:
	       U  is  nonsingular,  but	 RCOND is less than machine precision,
	       meaning that the matrix is singular to working precision.  Nev‐
	       ertheless,  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.

LAPACK version 3.0		 15 June 2000			     DPOSVX(l)
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