doubleGTcomputational(3) LAPACK doubleGTcomputational(3)

NAME¶

doubleGTcomputational

SYNOPSIS¶

Functions¶

subroutine dgtcon (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DGTCON subroutine dgtrfs (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGTRFS subroutine dgttrf (N, DL, D, DU, DU2, IPIV, INFO)
DGTTRF subroutine dgttrs (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
DGTTRS subroutine dgtts2 (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)
DGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

Detailed Description¶

This is the group of double computational functions for GT matrices

Function Documentation¶

subroutine dgtcon (character NORM, integer N, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( * ) DU2, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

DGTCON

Purpose:

DGTCON estimates the reciprocal of the condition number of a real
tridiagonal matrix A using the LU factorization as computed by
DGTTRF.
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 DOUBLE PRECISION array, dimension (N-1)
The (n-1) multipliers that define the matrix L from the
LU factorization of A as computed by DGTTRF.

D

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

DU

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

DU2

DU2 is DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION 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

NAG Ltd.

Date

December 2016

subroutine dgtrfs (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, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

DGTRFS

Purpose:

DGTRFS 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 DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of A.

D

D is DOUBLE PRECISION array, dimension (N)
The 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)
The (n-1) multipliers that define the matrix L from the
LU factorization of A as computed by DGTTRF.

DF

DF is DOUBLE PRECISION array, dimension (N)
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)
The (n-1) elements of the first superdiagonal of U.

DU2

DU2 is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DGTTRS.
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 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

Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

subroutine dgttrf (integer N, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( * ) DU2, integer, dimension( * ) IPIV, integer INFO)¶

DGTTRF

Purpose:

DGTTRF 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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

NAG Ltd.

Date

December 2016

subroutine dgttrs (character TRANS, integer N, integer NRHS, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( * ) DU2, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

DGTTRS

Purpose:

DGTTRS 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 DGTTRF.

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 DOUBLE PRECISION array, dimension (N-1)
The (n-1) multipliers that define the matrix L from the
LU factorization of A.

D

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

DU

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

DU2

DU2 is DOUBLE PRECISION 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 DOUBLE PRECISION 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

NAG Ltd.

Date

December 2016

subroutine dgtts2 (integer ITRANS, integer N, integer NRHS, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( * ) DU2, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB)¶

DGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

Purpose:

DGTTS2 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 DGTTRF.

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 DOUBLE PRECISION array, dimension (N-1)
The (n-1) multipliers that define the matrix L from the
LU factorization of A.

D

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

DU

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

DU2

DU2 is DOUBLE PRECISION 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 DOUBLE PRECISION 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