482 lines
18 KiB
FortranFixed
482 lines
18 KiB
FortranFixed
SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
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$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
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$ WORK, RWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER EQUED, FACT, TRANS
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INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
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DOUBLE PRECISION RCOND
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
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$ RWORK( * )
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COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
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$ WORK( * ), X( LDX, * )
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* ..
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*
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* Purpose
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* =======
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*
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* ZGESVX uses the LU factorization to compute the solution to a complex
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* system of linear equations
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* A * X = B,
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* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
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*
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* Error bounds on the solution and a condition estimate are also
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* provided.
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*
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* Description
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* ===========
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*
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* The following steps are performed:
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*
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* 1. If FACT = 'E', real scaling factors are computed to equilibrate
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* the system:
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* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
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* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
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* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
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* Whether or not the system will be equilibrated depends on the
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* scaling of the matrix A, but if equilibration is used, A is
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* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
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* or diag(C)*B (if TRANS = 'T' or 'C').
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*
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* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
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* matrix A (after equilibration if FACT = 'E') as
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* A = P * L * U,
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* where P is a permutation matrix, L is a unit lower triangular
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* matrix, and U is upper triangular.
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*
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* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
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* returns with INFO = i. Otherwise, the factored form of A is used
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* to estimate the condition number of the matrix A. If the
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* reciprocal of the condition number is less than machine precision,
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* INFO = N+1 is returned as a warning, but the routine still goes on
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* to solve for X and compute error bounds as described below.
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*
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* 4. The system of equations is solved for X using the factored form
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* of A.
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*
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* 5. Iterative refinement is applied to improve the computed solution
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* matrix and calculate error bounds and backward error estimates
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* for it.
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*
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* 6. If equilibration was used, the matrix X is premultiplied by
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* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
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* that it solves the original system before equilibration.
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*
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* Arguments
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* =========
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*
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* FACT (input) CHARACTER*1
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* Specifies whether or not the factored form of the matrix A is
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* supplied on entry, and if not, whether the matrix A should be
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* equilibrated before it is factored.
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* = 'F': On entry, AF and IPIV contain the factored form of A.
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* If EQUED is not 'N', the matrix A has been
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* equilibrated with scaling factors given by R and C.
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* A, AF, and IPIV are not modified.
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* = 'N': The matrix A will be copied to AF and factored.
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* = 'E': The matrix A will be equilibrated if necessary, then
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* copied to AF and factored.
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*
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* TRANS (input) CHARACTER*1
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* Specifies the form of the system of equations:
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* = 'N': A * X = B (No transpose)
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* = 'T': A**T * X = B (Transpose)
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* = 'C': A**H * X = B (Conjugate transpose)
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*
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* N (input) INTEGER
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* The number of linear equations, i.e., the order of the
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* matrix A. N >= 0.
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*
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* NRHS (input) INTEGER
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* The number of right hand sides, i.e., the number of columns
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* of the matrices B and X. NRHS >= 0.
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*
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* A (input/output) COMPLEX*16 array, dimension (LDA,N)
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* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
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* not 'N', then A must have been equilibrated by the scaling
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* factors in R and/or C. A is not modified if FACT = 'F' or
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* 'N', or if FACT = 'E' and EQUED = 'N' on exit.
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*
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* On exit, if EQUED .ne. 'N', A is scaled as follows:
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* EQUED = 'R': A := diag(R) * A
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* EQUED = 'C': A := A * diag(C)
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* EQUED = 'B': A := diag(R) * A * diag(C).
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,N).
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*
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* AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
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* If FACT = 'F', then AF is an input argument and on entry
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* contains the factors L and U from the factorization
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* A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
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* AF is the factored form of the equilibrated matrix A.
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*
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* If FACT = 'N', then AF is an output argument and on exit
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* returns the factors L and U from the factorization A = P*L*U
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* of the original matrix A.
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*
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* If FACT = 'E', then AF is an output argument and on exit
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* returns the factors L and U from the factorization A = P*L*U
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* of the equilibrated matrix A (see the description of A for
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* the form of the equilibrated matrix).
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*
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* LDAF (input) INTEGER
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* The leading dimension of the array AF. LDAF >= max(1,N).
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*
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* IPIV (input or output) INTEGER array, dimension (N)
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* If FACT = 'F', then IPIV is an input argument and on entry
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* contains the pivot indices from the factorization A = P*L*U
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* as computed by ZGETRF; row i of the matrix was interchanged
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* with row IPIV(i).
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*
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* If FACT = 'N', then IPIV is an output argument and on exit
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* contains the pivot indices from the factorization A = P*L*U
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* of the original matrix A.
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*
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* If FACT = 'E', then IPIV is an output argument and on exit
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* contains the pivot indices from the factorization A = P*L*U
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* of the equilibrated matrix A.
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*
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* EQUED (input or output) CHARACTER*1
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* Specifies the form of equilibration that was done.
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* = 'N': No equilibration (always true if FACT = 'N').
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* = 'R': Row equilibration, i.e., A has been premultiplied by
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* diag(R).
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* = 'C': Column equilibration, i.e., A has been postmultiplied
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* by diag(C).
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* = 'B': Both row and column equilibration, i.e., A has been
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* replaced by diag(R) * A * diag(C).
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* EQUED is an input argument if FACT = 'F'; otherwise, it is an
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* output argument.
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*
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* R (input or output) DOUBLE PRECISION array, dimension (N)
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* The row scale factors for A. If EQUED = 'R' or 'B', A is
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* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
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* is not accessed. R is an input argument if FACT = 'F';
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* otherwise, R is an output argument. If FACT = 'F' and
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* EQUED = 'R' or 'B', each element of R must be positive.
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*
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* C (input or output) DOUBLE PRECISION array, dimension (N)
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* The column scale factors for A. If EQUED = 'C' or 'B', A is
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* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
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* is not accessed. C is an input argument if FACT = 'F';
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* otherwise, C is an output argument. If FACT = 'F' and
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* EQUED = 'C' or 'B', each element of C must be positive.
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*
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* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
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* On entry, the N-by-NRHS right hand side matrix B.
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* On exit,
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* if EQUED = 'N', B is not modified;
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* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
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* diag(R)*B;
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* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
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* overwritten by diag(C)*B.
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*
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* LDB (input) INTEGER
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* The leading dimension of the array B. LDB >= max(1,N).
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*
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* X (output) COMPLEX*16 array, dimension (LDX,NRHS)
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* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
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* to the original system of equations. Note that A and B are
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* modified on exit if EQUED .ne. 'N', and the solution to the
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* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
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* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
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* and EQUED = 'R' or 'B'.
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*
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* LDX (input) INTEGER
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* The leading dimension of the array X. LDX >= max(1,N).
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*
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* RCOND (output) DOUBLE PRECISION
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* The estimate of the reciprocal condition number of the matrix
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* A after equilibration (if done). If RCOND is less than the
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* machine precision (in particular, if RCOND = 0), the matrix
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* is singular to working precision. This condition is
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* indicated by a return code of INFO > 0.
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*
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* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
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* The estimated forward error bound for each solution vector
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* X(j) (the j-th column of the solution matrix X).
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* If XTRUE is the true solution corresponding to X(j), FERR(j)
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* is an estimated upper bound for the magnitude of the largest
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* element in (X(j) - XTRUE) divided by the magnitude of the
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* largest element in X(j). The estimate is as reliable as
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* the estimate for RCOND, and is almost always a slight
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* overestimate of the true error.
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*
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* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
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* The componentwise relative backward error of each solution
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* vector X(j) (i.e., the smallest relative change in
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* any element of A or B that makes X(j) an exact solution).
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*
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* WORK (workspace) COMPLEX*16 array, dimension (2*N)
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*
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* RWORK (workspace/output) DOUBLE PRECISION array, dimension (2*N)
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* On exit, RWORK(1) contains the reciprocal pivot growth
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* factor norm(A)/norm(U). The "max absolute element" norm is
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* used. If RWORK(1) is much less than 1, then the stability
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* of the LU factorization of the (equilibrated) matrix A
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* could be poor. This also means that the solution X, condition
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* estimator RCOND, and forward error bound FERR could be
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* unreliable. If factorization fails with 0<INFO<=N, then
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* RWORK(1) contains the reciprocal pivot growth factor for the
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* leading INFO columns of A.
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -i, the i-th argument had an illegal value
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* > 0: if INFO = i, and i is
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* <= N: U(i,i) is exactly zero. The factorization has
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* been completed, but the factor U is exactly
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* singular, so the solution and error bounds
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* could not be computed. RCOND = 0 is returned.
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* = N+1: U is nonsingular, but RCOND is less than machine
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* precision, meaning that the matrix is singular
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* to working precision. Nevertheless, the
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* solution and error bounds are computed because
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* there are a number of situations where the
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* computed solution can be more accurate than the
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* value of RCOND would suggest.
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
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CHARACTER NORM
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INTEGER I, INFEQU, J
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DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
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$ ROWCND, RPVGRW, SMLNUM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, ZLANGE, ZLANTR
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EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANTR
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZGECON, ZGEEQU, ZGERFS, ZGETRF, ZGETRS,
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$ ZLACPY, ZLAQGE
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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NOFACT = LSAME( FACT, 'N' )
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EQUIL = LSAME( FACT, 'E' )
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NOTRAN = LSAME( TRANS, 'N' )
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IF( NOFACT .OR. EQUIL ) THEN
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EQUED = 'N'
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ROWEQU = .FALSE.
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COLEQU = .FALSE.
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ELSE
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ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
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COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
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SMLNUM = DLAMCH( 'Safe minimum' )
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BIGNUM = ONE / SMLNUM
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END IF
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*
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* Test the input parameters.
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*
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IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
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$ THEN
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INFO = -1
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ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
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$ LSAME( TRANS, 'C' ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -6
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ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
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$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
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INFO = -10
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ELSE
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IF( ROWEQU ) THEN
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RCMIN = BIGNUM
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RCMAX = ZERO
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DO 10 J = 1, N
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RCMIN = MIN( RCMIN, R( J ) )
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RCMAX = MAX( RCMAX, R( J ) )
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10 CONTINUE
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IF( RCMIN.LE.ZERO ) THEN
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INFO = -11
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ELSE IF( N.GT.0 ) THEN
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ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
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ELSE
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ROWCND = ONE
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END IF
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END IF
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IF( COLEQU .AND. INFO.EQ.0 ) THEN
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RCMIN = BIGNUM
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RCMAX = ZERO
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DO 20 J = 1, N
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RCMIN = MIN( RCMIN, C( J ) )
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RCMAX = MAX( RCMAX, C( J ) )
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20 CONTINUE
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IF( RCMIN.LE.ZERO ) THEN
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INFO = -12
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ELSE IF( N.GT.0 ) THEN
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COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
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ELSE
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COLCND = ONE
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -14
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ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
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INFO = -16
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END IF
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END IF
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZGESVX', -INFO )
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RETURN
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END IF
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*
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IF( EQUIL ) THEN
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*
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* Compute row and column scalings to equilibrate the matrix A.
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*
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CALL ZGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
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IF( INFEQU.EQ.0 ) THEN
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*
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* Equilibrate the matrix.
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*
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CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
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$ EQUED )
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ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
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COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
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END IF
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END IF
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*
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* Scale the right hand side.
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*
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IF( NOTRAN ) THEN
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IF( ROWEQU ) THEN
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DO 40 J = 1, NRHS
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DO 30 I = 1, N
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B( I, J ) = R( I )*B( I, J )
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30 CONTINUE
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40 CONTINUE
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END IF
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ELSE IF( COLEQU ) THEN
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DO 60 J = 1, NRHS
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DO 50 I = 1, N
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B( I, J ) = C( I )*B( I, J )
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50 CONTINUE
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60 CONTINUE
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END IF
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*
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IF( NOFACT .OR. EQUIL ) THEN
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*
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* Compute the LU factorization of A.
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*
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CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
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CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
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*
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* Return if INFO is non-zero.
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*
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IF( INFO.GT.0 ) THEN
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*
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* Compute the reciprocal pivot growth factor of the
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* leading rank-deficient INFO columns of A.
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*
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RPVGRW = ZLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
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$ RWORK )
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IF( RPVGRW.EQ.ZERO ) THEN
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RPVGRW = ONE
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ELSE
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RPVGRW = ZLANGE( 'M', N, INFO, A, LDA, RWORK ) /
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$ RPVGRW
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END IF
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RWORK( 1 ) = RPVGRW
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RCOND = ZERO
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RETURN
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END IF
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END IF
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*
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* Compute the norm of the matrix A and the
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* reciprocal pivot growth factor RPVGRW.
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*
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IF( NOTRAN ) THEN
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NORM = '1'
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ELSE
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NORM = 'I'
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END IF
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ANORM = ZLANGE( NORM, N, N, A, LDA, RWORK )
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RPVGRW = ZLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
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IF( RPVGRW.EQ.ZERO ) THEN
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RPVGRW = ONE
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ELSE
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RPVGRW = ZLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
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END IF
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*
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* Compute the reciprocal of the condition number of A.
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*
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CALL ZGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
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*
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* Compute the solution matrix X.
|
|
*
|
|
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
|
|
CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
|
|
*
|
|
* Use iterative refinement to improve the computed solution and
|
|
* compute error bounds and backward error estimates for it.
|
|
*
|
|
CALL ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
|
|
$ LDX, FERR, BERR, WORK, RWORK, INFO )
|
|
*
|
|
* Transform the solution matrix X to a solution of the original
|
|
* system.
|
|
*
|
|
IF( NOTRAN ) THEN
|
|
IF( COLEQU ) THEN
|
|
DO 80 J = 1, NRHS
|
|
DO 70 I = 1, N
|
|
X( I, J ) = C( I )*X( I, J )
|
|
70 CONTINUE
|
|
80 CONTINUE
|
|
DO 90 J = 1, NRHS
|
|
FERR( J ) = FERR( J ) / COLCND
|
|
90 CONTINUE
|
|
END IF
|
|
ELSE IF( ROWEQU ) THEN
|
|
DO 110 J = 1, NRHS
|
|
DO 100 I = 1, N
|
|
X( I, J ) = R( I )*X( I, J )
|
|
100 CONTINUE
|
|
110 CONTINUE
|
|
DO 120 J = 1, NRHS
|
|
FERR( J ) = FERR( J ) / ROWCND
|
|
120 CONTINUE
|
|
END IF
|
|
*
|
|
* Set INFO = N+1 if the matrix is singular to working precision.
|
|
*
|
|
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
|
|
$ INFO = N + 1
|
|
*
|
|
RWORK( 1 ) = RPVGRW
|
|
RETURN
|
|
*
|
|
* End of ZGESVX
|
|
*
|
|
END
|