SysTutorials

Linux Manual Pages › Section 3

cgbsvx-f(3)

cgbsvx.f –

Synopsis

Functions/Subroutines

subroutine cgbsvx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CGBSVX computes the solution to system of linear equations A * X = B for GB matrices

Function/Subroutine Documentation

subroutine cgbsvx (characterFACT, characterTRANS, integerN, integerKL, integerKU, integerNRHS, complex, dimension( ldab, * )AB, integerLDAB, complex, dimension( ldafb, * )AFB, integerLDAFB, integer, dimension( * )IPIV, characterEQUED, real, dimension( * )R, real, dimension( * )C, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, complex, dimension( * )WORK, real, dimension( * )RWORK, integerINFO)

CGBSVX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

 CGBSVX uses the LU factorization to compute the solution to a complex
 system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
 where A is a band matrix of order N with KL subdiagonals and KU
 superdiagonals, 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 by this subroutine:
 1. If FACT = 'E', real scaling factors are computed to equilibrate
 the system:
 TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
 TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
 TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
 or diag(C)*B (if TRANS = 'T' or 'C').
 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
 matrix A (after equilibration if FACT = 'E') as
 A = L * U,
 where L is a product of permutation and unit lower triangular
 matrices with KL subdiagonals, and U is upper triangular with
 KL+KU superdiagonals.
 3. If some U(i,i)=0, so that U 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.
 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of
 A. If EQUED is not 'N', the matrix A has been
 equilibrated with scaling factors given by R and C.
 AB, AFB, and IPIV are not 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.

TRANS

 TRANS is CHARACTER*1
 Specifies the form of the system of equations.
 = 'N': A * X = B (No transpose)
 = 'T': A**T * X = B (Transpose)
 = 'C': A**H * X = B (Conjugate transpose)

N

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

KL

 KL is INTEGER
 The number of subdiagonals within the band of A. KL >= 0.

KU

 KU is INTEGER
 The number of superdiagonals within the band of A. KU >= 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 matrix A in band storage, in rows 1 to KL+KU+1.
 The j-th column of A is stored in the j-th column of the
 array AB as follows:
 AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
 If FACT = 'F' and EQUED is not 'N', then A must have been
 equilibrated by the scaling factors in R and/or C. AB is not
 modified if FACT = 'F' or 'N', or if FACT = 'E' and
 EQUED = 'N' on exit.
 On exit, if EQUED .ne. 'N', A is scaled as follows:
 EQUED = 'R': A := diag(R) * A
 EQUED = 'C': A := A * diag(C)
 EQUED = 'B': A := diag(R) * A * diag(C).

LDAB

 LDAB is INTEGER
 The leading dimension of the array AB. LDAB >= KL+KU+1.

AFB

 AFB is COMPLEX array, dimension (LDAFB,N)
 If FACT = 'F', then AFB is an input argument and on entry
 contains details of the LU factorization of the band matrix
 A, as computed by CGBTRF. U is stored as an upper triangular
 band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
 and the multipliers used during the factorization are stored
 in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
 the factored form of the equilibrated matrix A.
 If FACT = 'N', then AFB is an output argument and on exit
 returns details of the LU factorization of A.
 If FACT = 'E', then AFB is an output argument and on exit
 returns details of the LU factorization of the equilibrated
 matrix A (see the description of AB for the form of the
 equilibrated matrix).

LDAFB

 LDAFB is INTEGER
 The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.

IPIV

 IPIV is INTEGER array, dimension (N)
 If FACT = 'F', then IPIV is an input argument and on entry
 contains the pivot indices from the factorization A = L*U
 as computed by CGBTRF; row i of the matrix was interchanged
 with row IPIV(i).
 If FACT = 'N', then IPIV is an output argument and on exit
 contains the pivot indices from the factorization A = L*U
 of the original matrix A.
 If FACT = 'E', then IPIV is an output argument and on exit
 contains the pivot indices from the factorization A = L*U
 of the equilibrated matrix A.

EQUED

 EQUED is CHARACTER*1
 Specifies the form of equilibration that was done.
 = 'N': No equilibration (always true if FACT = 'N').
 = 'R': Row equilibration, i.e., A has been premultiplied by
 diag(R).
 = 'C': Column equilibration, i.e., A has been postmultiplied
 by diag(C).
 = 'B': Both row and column equilibration, i.e., A has been
 replaced by diag(R) * A * diag(C).
 EQUED is an input argument if FACT = 'F'; otherwise, it is an
 output argument.

R

 R is REAL array, dimension (N)
 The row scale factors for A. If EQUED = 'R' or 'B', A is
 multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
 is not accessed. R is an input argument if FACT = 'F';
 otherwise, R is an output argument. If FACT = 'F' and
 EQUED = 'R' or 'B', each element of R must be positive.

C

 C is REAL array, dimension (N)
 The column scale factors for A. If EQUED = 'C' or 'B', A is
 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
 is not accessed. C is an input argument if FACT = 'F';
 otherwise, C is an output argument. If FACT = 'F' and
 EQUED = 'C' or 'B', each element of C must be positive.

B

 B is COMPLEX array, dimension (LDB,NRHS)
 On entry, the right hand side matrix B.
 On exit,
 if EQUED = 'N', B is not modified;
 if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
 diag(R)*B;
 if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
 overwritten by diag(C)*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 A and B are
 modified on exit if EQUED .ne. 'N', and the solution to the
 equilibrated system is inv(diag(C))*X if TRANS = 'N' and
 EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
 and EQUED = 'R' or 'B'.

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)
 On exit, RWORK(1) contains the reciprocal pivot growth
 factor norm(A)/norm(U). The "max absolute element" norm is
 used. If RWORK(1) is much less than 1, then the stability
 of the LU factorization of the (equilibrated) matrix A
 could be poor. This also means that the solution X, condition
 estimator RCOND, and forward error bound FERR could be
 unreliable. If factorization fails with 0<INFO<=N, then
 RWORK(1) contains the reciprocal pivot growth factor for the
 leading INFO columns of A.

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: U(i,i) is exactly zero. The factorization
 has been completed, but the factor U is exactly
 singular, so the solution and error bounds
 could not be 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

Definition at line 369 of file cgbsvx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.