dpbsvx.f

          SUBROUTINE DPBSVX( 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
          DOUBLE PRECISION   RCOND
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
         $                   BERR( * ), FERR( * ), S( * ), WORK( * ),
         $                   X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DPBSVX 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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 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) 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) DOUBLE PRECISION 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 ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            EQUIL, NOFACT, RCEQU, UPPER
          INTEGER            I, INFEQU, J, J1, J2
          DOUBLE PRECISION   AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, DLANSB
          EXTERNAL           LSAME, DLAMCH, DLANSB
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS,
         $                   DPBTRF, DPBTRS, 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 = 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.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( 'DPBSVX', -INFO )
             RETURN
          END IF
    *
          IF( EQUIL ) THEN
    *
    *        Compute row and column scalings to equilibrate the matrix A.
    *
             CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
             IF( INFEQU.EQ.0 ) THEN
    *
    *           Equilibrate the matrix.
    *
                CALL DLAQSB( 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 DCOPY( 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 DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
       50       CONTINUE
             END IF
    *
             CALL DPBTRF( 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 = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
    *
    *     Compute the reciprocal of the condition number of A.
    *
          CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK,
         $             INFO )
    *
    *     Compute the solution matrix X.
    *
          CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
          CALL DPBTRS( 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 DPBRFS( 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.DLAMCH( 'Epsilon' ) )
         $   INFO = N + 1
    *
          RETURN
    *
    *     End of DPBSVX
    *
          END