gtsvx(3)

NAME
gtsvx - gtsvx: factor and solve, expert

SYNOPSIS

  Functions
subroutine cgtsvx (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CGTSVX computes the solution to system of linear equations A * X = B for GT matrices
subroutine dgtsvx (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DGTSVX computes the solution to system of linear equations A * X = B for GT matrices
subroutine sgtsvx (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SGTSVX computes the solution to system of linear equations A * X = B for GT matrices
subroutine zgtsvx (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Detailed Description

Function Documentation

  subroutine cgtsvx (character fact, character trans, integer n, integer nrhs, complex, dimension( * ) dl, complex, dimension( * ) d, complex, dimension( * ) du, complex, dimension( * ) dlf, complex, dimension( * ) df, complex, dimension( * ) duf, complex, dimension( * ) du2, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

    !>
    !> CGTSVX 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 tridiagonal matrix of order N 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 = 'N', the LU decomposition is used to factor the matrix A
    !>    as A = L * U, where L is a product of permutation and unit lower
    !>    bidiagonal matrices and U is upper triangular with nonzeros in
    !>    only the main diagonal and first two superdiagonals.
    !>
    !> 2. 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.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored form
    !>                  of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
    !>                  be modified.
    !>          = 'N':  The matrix will be copied to DLF, DF, and DUF
    !>                  and factored.
    !>

    TRANS

    !>          TRANS is 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

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>          The number of right hand sides, i.e., the number of columns
    !>          of the matrix B.  NRHS >= 0.
    !>

    DL

    !>          DL is COMPLEX array, dimension (N-1)
    !>          The (n-1) subdiagonal elements of A.
    !>

    D

    !>          D is COMPLEX array, dimension (N)
    !>          The n diagonal elements of A.
    !>

    DU

    !>          DU is COMPLEX array, dimension (N-1)
    !>          The (n-1) superdiagonal elements of A.
    !>

    DLF

    !>          DLF is COMPLEX array, dimension (N-1)
    !>          If FACT = 'F', then DLF is an input argument and on entry
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A as computed by CGTTRF.
    !>
    !>          If FACT = 'N', then DLF is an output argument and on exit
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A.
    !>

    DF

    !>          DF is COMPLEX array, dimension (N)
    !>          If FACT = 'F', then DF is an input argument and on entry
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>
    !>          If FACT = 'N', then DF is an output argument and on exit
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>

    DUF

    !>          DUF is COMPLEX array, dimension (N-1)
    !>          If FACT = 'F', then DUF is an input argument and on entry
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>
    !>          If FACT = 'N', then DUF is an output argument and on exit
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>

    DU2

    !>          DU2 is COMPLEX array, dimension (N-2)
    !>          If FACT = 'F', then DU2 is an input argument and on entry
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>
    !>          If FACT = 'N', then DU2 is an output argument and on exit
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains the pivot indices from the LU factorization of A as
    !>          computed by CGTTRF.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains the pivot indices from the LU factorization of A;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
    !>          a row interchange was not required.
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= max(1,N).
    !>

    X

    !>          X is COMPLEX array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

    !>          LDX is INTEGER
    !>          The leading dimension of the array X.  LDX >= max(1,N).
    !>

    RCOND

    !>          RCOND is REAL
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  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

    !>          FERR is 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

    !>          BERR is 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

    !>          WORK is COMPLEX array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is REAL array, dimension (N)
    !>

    INFO

    !>          INFO is 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 not been completed unless i = N, 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 291 of file cgtsvx.f.

  subroutine dgtsvx (character fact, character trans, integer n, integer nrhs, double precision, dimension( * ) dl, double precision, dimension( * ) d, double precision, dimension( * ) du, double precision, dimension( * ) dlf, double precision, dimension( * ) df, double precision, dimension( * ) duf, double precision, dimension( * ) du2, integer, dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

    !>
    !> DGTSVX uses the LU factorization to compute the solution to a real
    !> system of linear equations A * X = B or A**T * X = B,
    !> where A is a tridiagonal matrix of order N 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 = 'N', the LU decomposition is used to factor the matrix A
    !>    as A = L * U, where L is a product of permutation and unit lower
    !>    bidiagonal matrices and U is upper triangular with nonzeros in
    !>    only the main diagonal and first two superdiagonals.
    !>
    !> 2. 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.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
    !>                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
    !>                  will not be modified.
    !>          = 'N':  The matrix will be copied to DLF, DF, and DUF
    !>                  and factored.
    !>

    TRANS

    !>          TRANS is 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 = Transpose)
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>          The number of right hand sides, i.e., the number of columns
    !>          of the matrix B.  NRHS >= 0.
    !>

    DL

    !>          DL is DOUBLE PRECISION array, dimension (N-1)
    !>          The (n-1) subdiagonal elements of A.
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          The n diagonal elements of A.
    !>

    DU

    !>          DU is DOUBLE PRECISION array, dimension (N-1)
    !>          The (n-1) superdiagonal elements of A.
    !>

    DLF

    !>          DLF is DOUBLE PRECISION array, dimension (N-1)
    !>          If FACT = 'F', then DLF is an input argument and on entry
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A as computed by DGTTRF.
    !>
    !>          If FACT = 'N', then DLF is an output argument and on exit
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A.
    !>

    DF

    !>          DF is DOUBLE PRECISION array, dimension (N)
    !>          If FACT = 'F', then DF is an input argument and on entry
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>
    !>          If FACT = 'N', then DF is an output argument and on exit
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>

    DUF

    !>          DUF is DOUBLE PRECISION array, dimension (N-1)
    !>          If FACT = 'F', then DUF is an input argument and on entry
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>
    !>          If FACT = 'N', then DUF is an output argument and on exit
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>

    DU2

    !>          DU2 is DOUBLE PRECISION array, dimension (N-2)
    !>          If FACT = 'F', then DU2 is an input argument and on entry
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>
    !>          If FACT = 'N', then DU2 is an output argument and on exit
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains the pivot indices from the LU factorization of A as
    !>          computed by DGTTRF.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains the pivot indices from the LU factorization of A;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
    !>          a row interchange was not required.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= max(1,N).
    !>

    X

    !>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

    !>          LDX is INTEGER
    !>          The leading dimension of the array X.  LDX >= max(1,N).
    !>

    RCOND

    !>          RCOND is DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  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

    !>          FERR is 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

    !>          BERR is 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

    !>          WORK is DOUBLE PRECISION array, dimension (3*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (N)
    !>

    INFO

    !>          INFO is 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 not been completed unless i = N, 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 290 of file dgtsvx.f.

  subroutine sgtsvx (character fact, character trans, integer n, integer nrhs, real, dimension( * ) dl, real, dimension( * ) d, real, dimension( * ) du, real, dimension( * ) dlf, real, dimension( * ) df, real, dimension( * ) duf, real, dimension( * ) du2, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

    !>
    !> SGTSVX uses the LU factorization to compute the solution to a real
    !> system of linear equations A * X = B or A**T * X = B,
    !> where A is a tridiagonal matrix of order N 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 = 'N', the LU decomposition is used to factor the matrix A
    !>    as A = L * U, where L is a product of permutation and unit lower
    !>    bidiagonal matrices and U is upper triangular with nonzeros in
    !>    only the main diagonal and first two superdiagonals.
    !>
    !> 2. 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.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
    !>                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
    !>                  will not be modified.
    !>          = 'N':  The matrix will be copied to DLF, DF, and DUF
    !>                  and factored.
    !>

    TRANS

    !>          TRANS is 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 = Transpose)
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>          The number of right hand sides, i.e., the number of columns
    !>          of the matrix B.  NRHS >= 0.
    !>

    DL

    !>          DL is REAL array, dimension (N-1)
    !>          The (n-1) subdiagonal elements of A.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          The n diagonal elements of A.
    !>

    DU

    !>          DU is REAL array, dimension (N-1)
    !>          The (n-1) superdiagonal elements of A.
    !>

    DLF

    !>          DLF is REAL array, dimension (N-1)
    !>          If FACT = 'F', then DLF is an input argument and on entry
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A as computed by SGTTRF.
    !>
    !>          If FACT = 'N', then DLF is an output argument and on exit
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A.
    !>

    DF

    !>          DF is REAL array, dimension (N)
    !>          If FACT = 'F', then DF is an input argument and on entry
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>
    !>          If FACT = 'N', then DF is an output argument and on exit
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>

    DUF

    !>          DUF is REAL array, dimension (N-1)
    !>          If FACT = 'F', then DUF is an input argument and on entry
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>
    !>          If FACT = 'N', then DUF is an output argument and on exit
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>

    DU2

    !>          DU2 is REAL array, dimension (N-2)
    !>          If FACT = 'F', then DU2 is an input argument and on entry
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>
    !>          If FACT = 'N', then DU2 is an output argument and on exit
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains the pivot indices from the LU factorization of A as
    !>          computed by SGTTRF.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains the pivot indices from the LU factorization of A;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
    !>          a row interchange was not required.
    !>

    B

    !>          B is REAL array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= max(1,N).
    !>

    X

    !>          X is REAL array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

    !>          LDX is INTEGER
    !>          The leading dimension of the array X.  LDX >= max(1,N).
    !>

    RCOND

    !>          RCOND is REAL
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  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

    !>          FERR is 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

    !>          BERR is 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

    !>          WORK is REAL array, dimension (3*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (N)
    !>

    INFO

    !>          INFO is 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 not been completed unless i = N, 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 290 of file sgtsvx.f.

  subroutine zgtsvx (character fact, character trans, integer n, integer nrhs, complex*16, dimension( * ) dl, complex*16, dimension( * ) d, complex*16, dimension( * ) du, complex*16, dimension( * ) dlf, complex*16, dimension( * ) df, complex*16, dimension( * ) duf, complex*16, dimension( * ) du2, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

    !>
    !> ZGTSVX 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 tridiagonal matrix of order N 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 = 'N', the LU decomposition is used to factor the matrix A
    !>    as A = L * U, where L is a product of permutation and unit lower
    !>    bidiagonal matrices and U is upper triangular with nonzeros in
    !>    only the main diagonal and first two superdiagonals.
    !>
    !> 2. 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.
    !>
    !> 3. The system of equations is solved for X using the factored form
    !>    of A.
    !>
    !> 4. Iterative refinement is applied to improve the computed solution
    !>    matrix and calculate error bounds and backward error estimates
    !>    for it.
    !>

Parameters
    FACT

    !>          FACT is CHARACTER*1
    !>          Specifies whether or not the factored form of A has been
    !>          supplied on entry.
    !>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored form
    !>                  of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
    !>                  be modified.
    !>          = 'N':  The matrix will be copied to DLF, DF, and DUF
    !>                  and factored.
    !>

    TRANS

    !>          TRANS is 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

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    NRHS

    !>          NRHS is INTEGER
    !>          The number of right hand sides, i.e., the number of columns
    !>          of the matrix B.  NRHS >= 0.
    !>

    DL

    !>          DL is COMPLEX*16 array, dimension (N-1)
    !>          The (n-1) subdiagonal elements of A.
    !>

    D

    !>          D is COMPLEX*16 array, dimension (N)
    !>          The n diagonal elements of A.
    !>

    DU

    !>          DU is COMPLEX*16 array, dimension (N-1)
    !>          The (n-1) superdiagonal elements of A.
    !>

    DLF

    !>          DLF is COMPLEX*16 array, dimension (N-1)
    !>          If FACT = 'F', then DLF is an input argument and on entry
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A as computed by ZGTTRF.
    !>
    !>          If FACT = 'N', then DLF is an output argument and on exit
    !>          contains the (n-1) multipliers that define the matrix L from
    !>          the LU factorization of A.
    !>

    DF

    !>          DF is COMPLEX*16 array, dimension (N)
    !>          If FACT = 'F', then DF is an input argument and on entry
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>
    !>          If FACT = 'N', then DF is an output argument and on exit
    !>          contains the n diagonal elements of the upper triangular
    !>          matrix U from the LU factorization of A.
    !>

    DUF

    !>          DUF is COMPLEX*16 array, dimension (N-1)
    !>          If FACT = 'F', then DUF is an input argument and on entry
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>
    !>          If FACT = 'N', then DUF is an output argument and on exit
    !>          contains the (n-1) elements of the first superdiagonal of U.
    !>

    DU2

    !>          DU2 is COMPLEX*16 array, dimension (N-2)
    !>          If FACT = 'F', then DU2 is an input argument and on entry
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>
    !>          If FACT = 'N', then DU2 is an output argument and on exit
    !>          contains the (n-2) elements of the second superdiagonal of
    !>          U.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          If FACT = 'F', then IPIV is an input argument and on entry
    !>          contains the pivot indices from the LU factorization of A as
    !>          computed by ZGTTRF.
    !>
    !>          If FACT = 'N', then IPIV is an output argument and on exit
    !>          contains the pivot indices from the LU factorization of A;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
    !>          a row interchange was not required.
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB,NRHS)
    !>          The N-by-NRHS right hand side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B.  LDB >= max(1,N).
    !>

    X

    !>          X is COMPLEX*16 array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
    !>

    LDX

    !>          LDX is INTEGER
    !>          The leading dimension of the array X.  LDX >= max(1,N).
    !>

    RCOND

    !>          RCOND is DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A.  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

    !>          FERR is 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

    !>          BERR is 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

    !>          WORK is COMPLEX*16 array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (N)
    !>

    INFO

    !>          INFO is 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 not been completed unless i = N, 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 291 of file zgtsvx.f.

Author

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