144c6b23bd
Added lapack routines for calculation of condition number and to fill out the QR factorization capability.
332 lines
10 KiB
Fortran
332 lines
10 KiB
Fortran
SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
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$ X, LDX, FERR, BERR, WORK, IWORK, INFO )
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*
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* -- LAPACK routine (version 3.0) --
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* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
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* Courant Institute, Argonne National Lab, and Rice University
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* September 30, 1994
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*
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* .. Scalar Arguments ..
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CHARACTER TRANS
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INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * ), IWORK( * )
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DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
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$ BERR( * ), FERR( * ), 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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* DGERFS improves the computed solution to a system of linear
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* equations and provides error bounds and backward error estimates for
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* the solution.
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*
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* Arguments
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* =========
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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 = Transpose)
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*
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* N (input) INTEGER
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* The order of the 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) DOUBLE PRECISION array, dimension (LDA,N)
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* The original N-by-N matrix A.
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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) DOUBLE PRECISION array, dimension (LDAF,N)
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* The factors L and U from the factorization A = P*L*U
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* as computed by DGETRF.
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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) INTEGER array, dimension (N)
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* The pivot indices from DGETRF; for 1<=i<=N, row i of the
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* matrix was interchanged with row IPIV(i).
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*
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* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
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* The right hand side matrix 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 (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
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* On entry, the solution matrix X, as computed by DGETRS.
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* On exit, the improved solution matrix X.
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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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* 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) DOUBLE PRECISION array, dimension (3*N)
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*
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* IWORK (workspace) INTEGER array, dimension (N)
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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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*
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* Internal Parameters
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* ===================
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*
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* ITMAX is the maximum number of steps of iterative refinement.
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*
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* =====================================================================
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*
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* .. Parameters ..
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INTEGER ITMAX
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PARAMETER ( ITMAX = 5 )
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DOUBLE PRECISION ZERO
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PARAMETER ( ZERO = 0.0D+0 )
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DOUBLE PRECISION ONE
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PARAMETER ( ONE = 1.0D+0 )
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DOUBLE PRECISION TWO
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PARAMETER ( TWO = 2.0D+0 )
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DOUBLE PRECISION THREE
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PARAMETER ( THREE = 3.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN
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CHARACTER TRANST
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INTEGER COUNT, I, J, K, KASE, NZ
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DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
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* ..
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* .. External Subroutines ..
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EXTERNAL DAXPY, DCOPY, DGEMV, DGETRS, DLACON, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH
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EXTERNAL LSAME, DLAMCH
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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NOTRAN = LSAME( TRANS, 'N' )
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IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
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$ LSAME( TRANS, 'C' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -10
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ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
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INFO = -12
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DGERFS', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
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DO 10 J = 1, NRHS
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FERR( J ) = ZERO
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BERR( J ) = ZERO
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10 CONTINUE
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RETURN
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END IF
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*
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IF( NOTRAN ) THEN
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TRANST = 'T'
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ELSE
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TRANST = 'N'
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END IF
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*
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* NZ = maximum number of nonzero elements in each row of A, plus 1
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*
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NZ = N + 1
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EPS = DLAMCH( 'Epsilon' )
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SAFMIN = DLAMCH( 'Safe minimum' )
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SAFE1 = NZ*SAFMIN
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SAFE2 = SAFE1 / EPS
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*
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* Do for each right hand side
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*
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DO 140 J = 1, NRHS
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*
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COUNT = 1
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LSTRES = THREE
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20 CONTINUE
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*
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* Loop until stopping criterion is satisfied.
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*
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* Compute residual R = B - op(A) * X,
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* where op(A) = A, A**T, or A**H, depending on TRANS.
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*
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CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
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CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
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$ WORK( N+1 ), 1 )
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*
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* Compute componentwise relative backward error from formula
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*
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* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
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*
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* where abs(Z) is the componentwise absolute value of the matrix
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* or vector Z. If the i-th component of the denominator is less
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* than SAFE2, then SAFE1 is added to the i-th components of the
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* numerator and denominator before dividing.
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*
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DO 30 I = 1, N
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WORK( I ) = ABS( B( I, J ) )
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30 CONTINUE
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*
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* Compute abs(op(A))*abs(X) + abs(B).
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*
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IF( NOTRAN ) THEN
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DO 50 K = 1, N
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XK = ABS( X( K, J ) )
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DO 40 I = 1, N
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WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
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40 CONTINUE
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50 CONTINUE
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ELSE
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DO 70 K = 1, N
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S = ZERO
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DO 60 I = 1, N
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S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
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60 CONTINUE
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WORK( K ) = WORK( K ) + S
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70 CONTINUE
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END IF
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S = ZERO
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DO 80 I = 1, N
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IF( WORK( I ).GT.SAFE2 ) THEN
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S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
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ELSE
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S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
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$ ( WORK( I )+SAFE1 ) )
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END IF
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80 CONTINUE
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BERR( J ) = S
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*
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* Test stopping criterion. Continue iterating if
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* 1) The residual BERR(J) is larger than machine epsilon, and
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* 2) BERR(J) decreased by at least a factor of 2 during the
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* last iteration, and
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* 3) At most ITMAX iterations tried.
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*
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IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
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$ COUNT.LE.ITMAX ) THEN
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*
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* Update solution and try again.
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*
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CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
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$ INFO )
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CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
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LSTRES = BERR( J )
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COUNT = COUNT + 1
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GO TO 20
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END IF
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*
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* Bound error from formula
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*
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* norm(X - XTRUE) / norm(X) .le. FERR =
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* norm( abs(inv(op(A)))*
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* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
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*
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* where
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* norm(Z) is the magnitude of the largest component of Z
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* inv(op(A)) is the inverse of op(A)
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* abs(Z) is the componentwise absolute value of the matrix or
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* vector Z
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* NZ is the maximum number of nonzeros in any row of A, plus 1
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* EPS is machine epsilon
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*
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* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
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* is incremented by SAFE1 if the i-th component of
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* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
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*
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* Use DLACON to estimate the infinity-norm of the matrix
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* inv(op(A)) * diag(W),
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* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
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*
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DO 90 I = 1, N
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IF( WORK( I ).GT.SAFE2 ) THEN
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WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
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ELSE
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WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
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END IF
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90 CONTINUE
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*
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KASE = 0
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100 CONTINUE
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CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
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$ KASE )
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IF( KASE.NE.0 ) THEN
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IF( KASE.EQ.1 ) THEN
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*
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* Multiply by diag(W)*inv(op(A)**T).
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*
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CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
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$ N, INFO )
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DO 110 I = 1, N
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WORK( N+I ) = WORK( I )*WORK( N+I )
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110 CONTINUE
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ELSE
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*
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* Multiply by inv(op(A))*diag(W).
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*
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DO 120 I = 1, N
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WORK( N+I ) = WORK( I )*WORK( N+I )
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120 CONTINUE
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CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
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$ INFO )
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END IF
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GO TO 100
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END IF
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*
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* Normalize error.
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*
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LSTRES = ZERO
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DO 130 I = 1, N
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LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
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130 CONTINUE
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IF( LSTRES.NE.ZERO )
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$ FERR( J ) = FERR( J ) / LSTRES
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*
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140 CONTINUE
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*
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RETURN
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*
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* End of DGERFS
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*
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END
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