Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - uses the LU factorization to compute the solution to a complex system of linear
       equations A * X = B, A**T * X = B, or A**H * X = B,

SYNOPSIS

       SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R,  C,
                          B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

           CHARACTER      EQUED, FACT, TRANS

           INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

           DOUBLE         PRECISION RCOND

           INTEGER        IPIV( * )

           DOUBLE         PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )

           COMPLEX*16     AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE

       ZGBSVX  uses  the  LU  factorization to compute the solution to a complex 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  (Conjugate 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) COMPLEX*16 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) COMPLEX*16 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 ZGBTRF.  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 ZGBTRF; 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 array, dimension (2*N)

        RWORK   (workspace/output) DOUBLE PRECISION array, dimension (N)
                On exit, RWORK(1) contains the reciprocal pivot growth
                factor norm(A)/norm(U). The "max absolute element" norm is
                used. If RWORK(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
                RWORK(1) contains the reciprocal pivot growth factor for the
                leading INFO columns of A.

        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.

 LAPACK driver routine (version 3.2)        April 2011                            ZGBSVX(3lapack)