SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, \$ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH. * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), \$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) * .. * * Purpose * ======= * * SPBRFS improves the computed solution to a system of linear * equations when the coefficient matrix is symmetric positive definite * and banded, and provides error bounds and backward error estimates * for the solution. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * 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) REAL array, dimension (LDAB,N) * The upper or lower triangle of the symmetric band matrix A, * stored in the first KD+1 rows of the array. 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). * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD+1. * * AFB (input) REAL array, dimension (LDAFB,N) * The triangular factor U or L from the Cholesky factorization * A = U**T*U or A = L*L**T of the band matrix A as computed by * SPBTRF, in the same storage format as A (see AB). * * LDAFB (input) INTEGER * The leading dimension of the array AFB. LDAFB >= KD+1. * * B (input) REAL array, dimension (LDB,NRHS) * The right hand side matrix B. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (input/output) REAL array, dimension (LDX,NRHS) * On entry, the solution matrix X, as computed by SPBTRS. * On exit, the improved solution matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * 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) REAL array, dimension (3*N) * * IWORK (workspace) INTEGER array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Internal Parameters * =================== * * ITMAX is the maximum number of steps of iterative refinement. * * ===================================================================== * * .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) REAL ONE PARAMETER ( ONE = 1.0E+0 ) REAL TWO PARAMETER ( TWO = 2.0E+0 ) REAL THREE PARAMETER ( THREE = 3.0E+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER COUNT, I, J, K, KASE, L, NZ REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Subroutines .. EXTERNAL SAXPY, SCOPY, SLACN2, SPBTRS, SSBMV, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH EXTERNAL LSAME, SLAMCH * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDAFB.LT.KD+1 ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SPBRFS', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF * * NZ = maximum number of nonzero elements in each row of A, plus 1 * NZ = MIN( N+1, 2*KD+2 ) EPS = SLAMCH( 'Epsilon' ) SAFMIN = SLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS * * Do for each right hand side * DO 140 J = 1, NRHS * COUNT = 1 LSTRES = THREE 20 CONTINUE * * Loop until stopping criterion is satisfied. * * Compute residual R = B - A * X * CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL SSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE, \$ WORK( N+1 ), 1 ) * * Compute componentwise relative backward error from formula * * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) * * where abs(Z) is the componentwise absolute value of the matrix * or vector Z. If the i-th component of the denominator is less * than SAFE2, then SAFE1 is added to the i-th components of the * numerator and denominator before dividing. * DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE * * Compute abs(A)*abs(X) + abs(B). * IF( UPPER ) THEN DO 50 K = 1, N S = ZERO XK = ABS( X( K, J ) ) L = KD + 1 - K DO 40 I = MAX( 1, K-KD ), K - 1 WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) ) 40 CONTINUE WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO XK = ABS( X( K, J ) ) WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK L = 1 - K DO 60 I = K + 1, MIN( N, K+KD ) WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) ) 60 CONTINUE WORK( K ) = WORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / \$ ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S * * Test stopping criterion. Continue iterating if * 1) The residual BERR(J) is larger than machine epsilon, and * 2) BERR(J) decreased by at least a factor of 2 during the * last iteration, and * 3) At most ITMAX iterations tried. * IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. \$ COUNT.LE.ITMAX ) THEN * * Update solution and try again. * CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, \$ INFO ) CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF * * Bound error from formula * * norm(X - XTRUE) / norm(X) .le. FERR = * norm( abs(inv(A))* * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) * * where * norm(Z) is the magnitude of the largest component of Z * inv(A) is the inverse of A * abs(Z) is the componentwise absolute value of the matrix or * vector Z * NZ is the maximum number of nonzeros in any row of A, plus 1 * EPS is machine epsilon * * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) * is incremented by SAFE1 if the i-th component of * abs(A)*abs(X) + abs(B) is less than SAFE2. * * Use SLACN2 to estimate the infinity-norm of the matrix * inv(A) * diag(W), * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) * DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE * KASE = 0 100 CONTINUE CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), \$ KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Multiply by diag(W)*inv(A**T). * CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, \$ INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( N+I )*WORK( I ) 110 CONTINUE ELSE IF( KASE.EQ.2 ) THEN * * Multiply by inv(A)*diag(W). * DO 120 I = 1, N WORK( N+I ) = WORK( N+I )*WORK( I ) 120 CONTINUE CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, \$ INFO ) END IF GO TO 100 END IF * * Normalize error. * LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) \$ FERR( J ) = FERR( J ) / LSTRES * 140 CONTINUE * RETURN * * End of SPBRFS * END