SUBROUTINE SPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, $ EQUED, S, 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, UPLO INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS REAL RCOND * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ BERR( * ), FERR( * ), S( * ), WORK( * ), $ X( LDX, * ) * .. * * Purpose * ======= * * SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to * compute the solution to a real system of linear equations * A * X = B, * where A is an N-by-N symmetric positive definite band matrix 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: * * 1. If FACT = 'E', real scaling factors are computed to equilibrate * the system: * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B. * * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to * factor the matrix A (after equilibration if FACT = 'E') as * A = U**T * U, if UPLO = 'U', or * A = L * L**T, if UPLO = 'L', * where U is an upper triangular band matrix, and L is a lower * triangular band matrix. * * 3. If the leading i-by-i principal minor is not positive definite, * 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(S) 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 contains the factored form of A. * If EQUED = 'Y', the matrix A has been equilibrated * with scaling factors given by S. AB and AFB will not * be 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. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The number of linear equations, i.e., 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/output) REAL array, dimension (LDAB,N) * On entry, the upper or lower triangle of the symmetric band * matrix A, stored in the first KD+1 rows of the array, except * if FACT = 'F' and EQUED = 'Y', then A must contain the * equilibrated matrix diag(S)*A*diag(S). 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). * See below for further details. * * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by * diag(S)*A*diag(S). * * LDAB (input) INTEGER * The leading dimension of the array A. LDAB >= KD+1. * * AFB (input or output) REAL array, dimension (LDAFB,N) * If FACT = 'F', then AFB is an input argument and on entry * contains the triangular factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T of the band matrix * A, in the same storage format as A (see AB). If EQUED = 'Y', * then AFB is the factored form of the equilibrated matrix A. * * If FACT = 'N', then AFB is an output argument and on exit * returns the triangular factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T. * * If FACT = 'E', then AFB is an output argument and on exit * returns the triangular factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T of the equilibrated * matrix A (see the description of A for the form of the * equilibrated matrix). * * LDAFB (input) INTEGER * The leading dimension of the array AFB. LDAFB >= KD+1. * * EQUED (input or output) CHARACTER*1 * Specifies the form of equilibration that was done. * = 'N': No equilibration (always true if FACT = 'N'). * = 'Y': Equilibration was done, i.e., A has been replaced by * diag(S) * A * diag(S). * EQUED is an input argument if FACT = 'F'; otherwise, it is an * output argument. * * S (input or output) REAL array, dimension (N) * The scale factors for A; not accessed if EQUED = 'N'. S is * an input argument if FACT = 'F'; otherwise, S is an output * argument. If FACT = 'F' and EQUED = 'Y', each element of S * must be positive. * * B (input/output) REAL array, dimension (LDB,NRHS) * On entry, the N-by-NRHS right hand side matrix B. * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', * B is overwritten by diag(S) * B. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (output) REAL 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 if EQUED = 'Y', * A and B are modified on exit, and the solution to the * equilibrated system is inv(diag(S))*X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * RCOND (output) 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 (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 * > 0: if INFO = i, and i is * <= N: the leading minor of order i of A is * not positive definite, so the factorization * could not be completed, and the solution has not * been 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. * * Further Details * =============== * * The band storage scheme is illustrated by the following example, when * N = 6, KD = 2, and UPLO = 'U': * * Two-dimensional storage of the symmetric matrix A: * * a11 a12 a13 * a22 a23 a24 * a33 a34 a35 * a44 a45 a46 * a55 a56 * (aij=conjg(aji)) a66 * * Band storage of the upper triangle of A: * * * * a13 a24 a35 a46 * * a12 a23 a34 a45 a56 * a11 a22 a33 a44 a55 a66 * * Similarly, if UPLO = 'L' the format of A is as follows: * * a11 a22 a33 a44 a55 a66 * a21 a32 a43 a54 a65 * * a31 a42 a53 a64 * * * * Array elements marked * are not used by the routine. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL EQUIL, NOFACT, RCEQU, UPPER INTEGER I, INFEQU, J, J1, J2 REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH, SLANSB EXTERNAL LSAME, SLAMCH, SLANSB * .. * .. External Subroutines .. EXTERNAL SCOPY, SLACPY, SLAQSB, SPBCON, SPBEQU, SPBRFS, $ SPBTRF, SPBTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) UPPER = LSAME( UPLO, 'U' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' RCEQU = .FALSE. ELSE RCEQU = LSAME( EQUED, 'Y' ) SMLNUM = SLAMCH( '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.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -7 ELSE IF( LDAFB.LT.KD+1 ) THEN INFO = -9 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -10 ELSE IF( RCEQU ) THEN SMIN = BIGNUM SMAX = ZERO DO 10 J = 1, N SMIN = MIN( SMIN, S( J ) ) SMAX = MAX( SMAX, S( J ) ) 10 CONTINUE IF( SMIN.LE.ZERO ) THEN INFO = -11 ELSE IF( N.GT.0 ) THEN SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) ELSE SCOND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -13 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -15 END IF END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SPBSVX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL SLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED ) RCEQU = LSAME( EQUED, 'Y' ) END IF END IF * * Scale the right-hand side. * IF( RCEQU ) THEN DO 30 J = 1, NRHS DO 20 I = 1, N B( I, J ) = S( I )*B( I, J ) 20 CONTINUE 30 CONTINUE END IF * IF( NOFACT .OR. EQUIL ) THEN * * Compute the Cholesky factorization A = U'*U or A = L*L'. * IF( UPPER ) THEN DO 40 J = 1, N J1 = MAX( J-KD, 1 ) CALL SCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1, $ AFB( KD+1-J+J1, J ), 1 ) 40 CONTINUE ELSE DO 50 J = 1, N J2 = MIN( J+KD, N ) CALL SCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 ) 50 CONTINUE END IF * CALL SPBTRF( UPLO, N, KD, AFB, LDAFB, INFO ) * * Return if INFO is non-zero. * IF( INFO.GT.0 )THEN RCOND = ZERO RETURN END IF END IF * * Compute the norm of the matrix A. * ANORM = SLANSB( '1', UPLO, N, KD, AB, LDAB, WORK ) * * Compute the reciprocal of the condition number of A. * CALL SPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK, $ INFO ) * * Compute the solution matrix X. * CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL SPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO ) * * Use iterative refinement to improve the computed solution and * compute error bounds and backward error estimates for it. * CALL SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, $ LDX, FERR, BERR, WORK, IWORK, INFO ) * * Transform the solution matrix X to a solution of the original * system. * IF( RCEQU ) THEN DO 70 J = 1, NRHS DO 60 I = 1, N X( I, J ) = S( I )*X( I, J ) 60 CONTINUE 70 CONTINUE DO 80 J = 1, NRHS FERR( J ) = FERR( J ) / SCOND 80 CONTINUE END IF * * Set INFO = N+1 if the matrix is singular to working precision. * IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) $ INFO = N + 1 * RETURN * * End of SPBSVX * END