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cpbsvx-f(3)

cpbsvx.f –

Synopsis

Functions/Subroutines

subroutine cpbsvx (FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Function/Subroutine Documentation

subroutine cpbsvx (characterFACT, characterUPLO, integerN, integerKD, integerNRHS, complex, dimension( ldab, * )AB, integerLDAB, complex, dimension( ldafb, * )AFB, integerLDAFB, characterEQUED, real, dimension( * )S, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, complex, dimension( * )WORK, real, dimension( * )RWORK, integerINFO)

CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

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. 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 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).

LDAB

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

AFB

 AFB is 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

 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 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

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

X

 X is COMPLEX 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 COMPLEX array, dimension (2*N)

RWORK

 RWORK is REAL 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 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:
 * * 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 341 of file cpbsvx.f.

Author

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