CPBSVX man page on IRIX

Man page or keyword search:  
man Server   31559 pages
apropos Keyword Search (all sections)
Output format
IRIX logo
[printable version]



CPBSVX(3F)							    CPBSVX(3F)

NAME
     CPBSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to
     compute the solution to a complex system of linear equations  A * X = B,

SYNOPSIS
     SUBROUTINE CPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED,
			S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
			INFO )

	 CHARACTER	EQUED, FACT, UPLO

	 INTEGER	INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS

	 REAL		RCOND

	 REAL		BERR( * ), FERR( * ), RWORK( * ), S( * )

	 COMPLEX	AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
			), X( LDX, * )

PURPOSE
     CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
     compute the solution to a complex system of linear equations
	A * X = B, where A is an N-by-N Hermitian 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**H * U,  if UPLO = 'U', or
	   A = L * L**H,  if UPLO = 'L',
	where U is an upper triangular band matrix, and L is a lower
	triangular band matrix.

     3. 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, steps 4-6 are skipped.

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

									Page 1

CPBSVX(3F)							    CPBSVX(3F)

     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, 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    (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.

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

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

     AB	     (input/output) COMPLEX array, dimension (LDAB,N)
	     On entry, the upper or lower triangle of the Hermitian 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).

									Page 2

CPBSVX(3F)							    CPBSVX(3F)

     LDAB    (input) INTEGER
	     The leading dimension of the array A.  LDAB >= KD+1.

     AFB     (input or output) COMPLEX 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**H*U or A = L*L**H 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**H*U or A = L*L**H.

	     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**H*U or A = L*L**H of the equilibrated matrix A (see the
	     description of A for the form of the equilibrated matrix).

     LDAFB   (input) INTEGER
	     The leading dimension of the array AFB.  LDAFB >= KD+1.

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

     X	     (output) COMPLEX array, dimension (LDX,NRHS)
	     If INFO = 0, 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     (input) INTEGER
	     The leading dimension of the array X.  LDX >= max(1,N).

									Page 3

CPBSVX(3F)							    CPBSVX(3F)

     RCOND   (output) 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, and the solution and error bounds are not
	     computed.

     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
	     estimate is as reliable as the estimate for RCOND, and is almost
	     always a slight overestimate 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 solution).

     WORK    (workspace) COMPLEX array, dimension (2*N)

     RWORK   (workspace) REAL 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 definite,
	     so the factorization could not be completed, and the solution has
	     not been computed.	 = N+1: RCOND is less than machine precision.
	     The factorization has been completed, but the matrix is singular
	     to working precision, and the solution and error bounds have not
	     been computed.

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 Hermitian 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:

									Page 4

CPBSVX(3F)							    CPBSVX(3F)

	 *    *	  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.

									Page 5

[top]

List of man pages available for IRIX

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

[legal] [privacy] [GNU] [policy] [cookies] [netiquette] [sponsors] [FAQ]
Tweet
Polarhome, production since 1999.
Member of Polarhome portal.
Based on Fawad Halim's script.
....................................................................
Vote for polarhome
Free Shell Accounts :: the biggest list on the net