Functions

subroutine sgtcon (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGTCON
subroutine sgtrfs (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGTRFS
subroutine sgttrf (N, DL, D, DU, DU2, IPIV, INFO)
SGTTRF
subroutine sgttrs (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
SGTTRS
subroutine sgtts2 (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)
SGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

Detailed Description

This is the group of real computational functions for GT matrices

Function Documentation

subroutine sgtcon (character NORM, integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SGTCON

Purpose:

 SGTCON estimates the reciprocal of the condition number of a real
 tridiagonal matrix A using the LU factorization as computed by
 SGTTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

 

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

N

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

DL

          DL is REAL array, dimension (N-1)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A as computed by SGTTRF.

D

          D is REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)
          The (n-1) elements of the first superdiagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)
          The (n-2) elements of the second superdiagonal of U.

IPIV

          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, 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.

ANORM

          ANORM is REAL
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is REAL array, dimension (2*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

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine sgtrfs (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, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SGTRFS

Purpose:

 SGTRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is tridiagonal, and provides
 error bounds and backward error estimates for the solution.

 

Parameters:

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 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)
          The (n-1) multipliers that define the matrix L from the
          LU factorization of A as computed by SGTTRF.

DF

          DF is REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

DUF

          DUF is REAL array, dimension (N-1)
          The (n-1) elements of the first superdiagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)
          The (n-2) elements of the second superdiagonal of U.

IPIV

          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, 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 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)
          On entry, the solution matrix X, as computed by SGTTRS.
          On exit, the improved solution matrix X.

LDX

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

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

 

Internal Parameters:

  ITMAX is the maximum number of steps of iterative refinement.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine sgttrf (integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, integer INFO)

SGTTRF

Purpose:

 SGTTRF computes an LU factorization of a real tridiagonal matrix A
 using elimination with partial pivoting and row interchanges.
 The factorization has the form
    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.

 

Parameters:

N

          N is INTEGER
          The order of the matrix A.

DL

          DL is REAL array, dimension (N-1)
          On entry, DL must contain the (n-1) sub-diagonal elements of
          A.
          On exit, DL is overwritten by the (n-1) multipliers that
          define the matrix L from the LU factorization of A.

D

          D is REAL array, dimension (N)
          On entry, D must contain the diagonal elements of A.
          On exit, D is overwritten by the n diagonal elements of the
          upper triangular matrix U from the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)
          On entry, DU must contain the (n-1) super-diagonal elements
          of A.
          On exit, DU is overwritten by the (n-1) elements of the first
          super-diagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)
          On exit, DU2 is overwritten by the (n-2) elements of the
          second super-diagonal of U.

IPIV

          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, 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.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
          > 0:  if INFO = k, U(k,k) is exactly zero. The factorization
                has been completed, but the factor U is exactly
                singular, and division by zero will occur if it is used
                to solve a system of equations.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine sgttrs (character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SGTTRS

Purpose:

 SGTTRS solves one of the systems of equations
    A*X = B  or  A**T*X = B,
 with a tridiagonal matrix A using the LU factorization computed
 by SGTTRF.

 

Parameters:

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**T* X = B  (Conjugate transpose = Transpose)

N

          N is INTEGER
          The order of the matrix A.

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) multipliers that define the matrix L from the
          LU factorization of A.

D

          D is REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)
          The (n-1) elements of the first super-diagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)
          The (n-2) elements of the second super-diagonal of U.

IPIV

          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, 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)
          On entry, the matrix of right hand side vectors B.
          On exit, B is overwritten by the solution vectors X.

LDB

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

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine sgtts2 (integer ITRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB)

SGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

Purpose:

 SGTTS2 solves one of the systems of equations
    A*X = B  or  A**T*X = B,
 with a tridiagonal matrix A using the LU factorization computed
 by SGTTRF.

 

Parameters:

ITRANS

          ITRANS is INTEGER
          Specifies the form of the system of equations.
          = 0:  A * X = B  (No transpose)
          = 1:  A**T* X = B  (Transpose)
          = 2:  A**T* X = B  (Conjugate transpose = Transpose)

N

          N is INTEGER
          The order of the matrix A.

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) multipliers that define the matrix L from the
          LU factorization of A.

D

          D is REAL array, dimension (N)
          The n diagonal elements of the upper triangular matrix U from
          the LU factorization of A.

DU

          DU is REAL array, dimension (N-1)
          The (n-1) elements of the first super-diagonal of U.

DU2

          DU2 is REAL array, dimension (N-2)
          The (n-2) elements of the second super-diagonal of U.

IPIV

          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= n, 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)
          On entry, the matrix of right hand side vectors B.
          On exit, B is overwritten by the solution vectors X.

LDB

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

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Author

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