SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, $ RCOND, FERR, BERR, WORK, IWORK, INFO ) * * -- LAPACK driver routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER EQUED, FACT, TRANS INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND * .. * .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ BERR( * ), C( * ), FERR( * ), R( * ), $ WORK( * ), X( LDX, * ) * .. * * Purpose * ======= * * DGBSVX uses the LU factorization to compute the solution to a real * 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. * * Arguments * ========= * * FACT (input) 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 (input) 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 (Transpose) * * N (input) INTEGER * The number of linear equations, i.e., the order of the * matrix A. N >= 0. * * KL (input) INTEGER * The number of subdiagonals within the band of A. KL >= 0. * * KU (input) INTEGER * The number of superdiagonals within the band of A. KU >= 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/output) DOUBLE PRECISION 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 (input) INTEGER * The leading dimension of the array AB. LDAB >= KL+KU+1. * * AFB (input or output) DOUBLE PRECISION 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 DGBTRF. 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 (input) INTEGER * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. * * IPIV (input or output) 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 DGBTRF; 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 (input or output) 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 (input or output) DOUBLE PRECISION 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 (input or output) DOUBLE PRECISION 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 (input/output) DOUBLE PRECISION 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 (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (output) DOUBLE PRECISION 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 (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * RCOND (output) DOUBLE PRECISION * 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 (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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/output) DOUBLE PRECISION array, dimension (3*N) * On exit, WORK(1) contains the reciprocal pivot growth * factor norm(A)/norm(U). The "max absolute element" norm is * used. If WORK(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 * WORK(1) contains the reciprocal pivot growth factor for the * leading INFO columns of A. * * IWORK (workspace) INTEGER array, dimension (N) * * INFO (output) 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. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU CHARACTER NORM INTEGER I, INFEQU, J, J1, J2 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, $ ROWCND, RPVGRW, SMLNUM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGB, DLANTB EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB * .. * .. External Subroutines .. EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS, $ DLACPY, DLAQGB, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. Executable Statements .. * INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) NOTRAN = LSAME( TRANS, 'N' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' ROWEQU = .FALSE. COLEQU = .FALSE. ELSE ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF * * Test the input parameters. * IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) $ THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. $ LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KL.LT.0 ) THEN INFO = -4 ELSE IF( KU.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -8 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -10 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -12 ELSE IF( ROWEQU ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 10 J = 1, N RCMIN = MIN( RCMIN, R( J ) ) RCMAX = MAX( RCMAX, R( J ) ) 10 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -13 ELSE IF( N.GT.0 ) THEN ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE ROWCND = ONE END IF END IF IF( COLEQU .AND. INFO.EQ.0 ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 20 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 20 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -14 ELSE IF( N.GT.0 ) THEN COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE COLCND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -16 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -18 END IF END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBSVX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, EQUED ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF END IF * * Scale the right hand side. * IF( NOTRAN ) THEN IF( ROWEQU ) THEN DO 40 J = 1, NRHS DO 30 I = 1, N B( I, J ) = R( I )*B( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( COLEQU ) THEN DO 60 J = 1, NRHS DO 50 I = 1, N B( I, J ) = C( I )*B( I, J ) 50 CONTINUE 60 CONTINUE END IF * IF( NOFACT .OR. EQUIL ) THEN * * Compute the LU factorization of the band matrix A. * DO 70 J = 1, N J1 = MAX( J-KU, 1 ) J2 = MIN( J+KL, N ) CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1, $ AFB( KL+KU+1-J+J1, J ), 1 ) 70 CONTINUE * CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) * * Return if INFO is non-zero. * IF( INFO.GT.0 ) THEN * * Compute the reciprocal pivot growth factor of the * leading rank-deficient INFO columns of A. * ANORM = ZERO DO 90 J = 1, INFO DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 ) ANORM = MAX( ANORM, ABS( AB( I, J ) ) ) 80 CONTINUE 90 CONTINUE RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ), $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB, $ WORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = ANORM / RPVGRW END IF WORK( 1 ) = RPVGRW RCOND = ZERO RETURN END IF END IF * * Compute the norm of the matrix A and the * reciprocal pivot growth factor RPVGRW. * IF( NOTRAN ) THEN NORM = '1' ELSE NORM = 'I' END IF ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK ) RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW END IF * * Compute the reciprocal of the condition number of A. * CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND, $ WORK, IWORK, INFO ) * * Compute the solution matrix X. * CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, $ INFO ) * * Use iterative refinement to improve the computed solution and * compute error bounds and backward error estimates for it. * CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) * * Transform the solution matrix X to a solution of the original * system. * IF( NOTRAN ) THEN IF( COLEQU ) THEN DO 110 J = 1, NRHS DO 100 I = 1, N X( I, J ) = C( I )*X( I, J ) 100 CONTINUE 110 CONTINUE DO 120 J = 1, NRHS FERR( J ) = FERR( J ) / COLCND 120 CONTINUE END IF ELSE IF( ROWEQU ) THEN DO 140 J = 1, NRHS DO 130 I = 1, N X( I, J ) = R( I )*X( I, J ) 130 CONTINUE 140 CONTINUE DO 150 J = 1, NRHS FERR( J ) = FERR( J ) / ROWCND 150 CONTINUE END IF * * Set INFO = N+1 if the matrix is singular to working precision. * IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) $ INFO = N + 1 * WORK( 1 ) = RPVGRW RETURN * * End of DGBSVX * END