CentOS 7.0 - man page for spbsvx (centos section 3)

Linux & Unix Commands - Search Man Pages

Man Page or Keyword Search:   man
Select Man Page Set:       apropos Keyword Search (sections above)


spbsvx.f(3)				      LAPACK				      spbsvx.f(3)

NAME
       spbsvx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine spbsvx (FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X,
	   LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
	    SPBSVX computes the solution to system of linear equations A * X = B for OTHER
	   matrices

Function/Subroutine Documentation
   subroutine spbsvx (characterFACT, characterUPLO, integerN, integerKD, integerNRHS, real,
       dimension( ldab, * )AB, integerLDAB, real, dimension( ldafb, * )AFB, integerLDAFB,
       characterEQUED, real, dimension( * )S, real, dimension( ldb, * )B, integerLDB, real,
       dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( *
       )BERR, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
	SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

       Purpose:

	    SPBSVX uses 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,
	    where A is an N-by-N symmetric positive definite band matrix 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 = '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 band matrix, and L is a lower
	       triangular band 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.

       Parameters:
	   FACT

		     FACT is 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, AFB contains the factored form of A.
			     If EQUED = 'Y', the matrix A has been equilibrated
			     with scaling factors given by S.  AB and AFB will not
			     be modified.
		     = 'N':  The matrix A will be copied to AFB and factored.
		     = 'E':  The matrix A will be equilibrated if necessary, then
			     copied to AFB and factored.

	   UPLO

		     UPLO is CHARACTER*1
		     = 'U':  Upper triangle of A is stored;
		     = 'L':  Lower triangle of A is stored.

	   N

		     N is INTEGER
		     The number of linear equations, i.e., the order of the
		     matrix A.	N >= 0.

	   KD

		     KD is INTEGER
		     The number of superdiagonals of the matrix A if UPLO = 'U',
		     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

	   NRHS

		     NRHS is INTEGER
		     The number of right-hand sides, i.e., the number of columns
		     of the matrices B and X.  NRHS >= 0.

	   AB

		     AB is REAL array, dimension (LDAB,N)
		     On entry, the upper or lower triangle of the symmetric band
		     matrix A, stored in the first KD+1 rows of the array, except
		     if FACT = 'F' and EQUED = 'Y', then A must contain the
		     equilibrated matrix diag(S)*A*diag(S).  The j-th column of A
		     is stored in the j-th column of the array AB as follows:
		     if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
		     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
		     See below for further details.

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

	   LDAB

		     LDAB is INTEGER
		     The leading dimension of the array A.  LDAB >= KD+1.

	   AFB

		     AFB is REAL array, dimension (LDAFB,N)
		     If FACT = 'F', then AFB is an input argument and on entry
		     contains the triangular factor U or L from the Cholesky
		     factorization A = U**T*U or A = L*L**T of the band matrix
		     A, in the same storage format as A (see AB).  If EQUED = 'Y',
		     then AFB is the factored form of the equilibrated matrix A.

		     If FACT = 'N', then AFB is an output argument and on exit
		     returns the triangular factor U or L from the Cholesky
		     factorization A = U**T*U or A = L*L**T.

		     If FACT = 'E', then AFB is an output argument and on exit
		     returns the triangular factor U or L from the Cholesky
		     factorization 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).

	   LDAFB

		     LDAFB is INTEGER
		     The leading dimension of the array AFB.  LDAFB >= KD+1.

	   EQUED

		     EQUED is 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

		     S is REAL 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
		     argument.	If FACT = 'F' and EQUED = 'Y', each element of S
		     must be positive.

	   B

		     B is REAL 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 overwritten by diag(S) * B.

	   LDB

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

	   X

		     X is REAL 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
		     equilibrated system is inv(diag(S))*X.

	   LDX

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

	   RCOND

		     RCOND is REAL
		     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

		     FERR is 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 estimate is as reliable as
		     the estimate for RCOND, and is almost always a slight
		     overestimate of the true error.

	   BERR

		     BERR is 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 solution).

	   WORK

		     WORK is REAL array, dimension (3*N)

	   IWORK

		     IWORK is INTEGER array, dimension (N)

	   INFO

		     INFO is 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 definite, 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.  Nevertheless, 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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       Further Details:

	     The band storage scheme is illustrated by the following example, when
	     N = 6, KD = 2, and UPLO = 'U':

	     Two-dimensional storage of the symmetric matrix A:

		a11  a12  a13
		     a22  a23  a24
			  a33  a34  a35
			       a44  a45  a46
				    a55  a56
		(aij=conjg(aji))	 a66

	     Band storage of the upper triangle of A:

		 *    *   a13  a24  a35  a46
		 *   a12  a23  a34  a45  a56
		a11  a22  a33  a44  a55  a66

	     Similarly, if UPLO = 'L' the format of A is as follows:

		a11  a22  a33  a44  a55  a66
		a21  a32  a43  a54  a65   *
		a31  a42  a53  a64   *	  *

	     Array elements marked * are not used by the routine.

       Definition at line 342 of file spbsvx.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      spbsvx.f(3)
Unix & Linux Commands & Man Pages : ©2000 - 2018 Unix and Linux Forums


All times are GMT -4. The time now is 10:57 PM.

Unix & Linux Forums Content Copyright©1993-2018. All Rights Reserved.
×
UNIX.COM Login
Username:
Password:  
Show Password





Not a Forum Member?
Forgot Password?