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Manual

gtsvx

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

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

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 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 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 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.

subroutine dgtsvx (character fact, character trans, integer n, integernrhs, double precision, dimension( * ) dl, double precision, dimension(* ) d, double precision, dimension( * ) du, double precision,dimension( * ) dlf, double precision, dimension( * ) df, doubleprecision, 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, doubleprecision, 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:

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

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

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 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.

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 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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine sgtsvx (character fact, character trans, integer n, integernrhs, 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:

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

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

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 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.

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 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

FERR

BERR

WORK

WORK is REAL array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zgtsvx (character fact, character trans, integer n, integernrhs, complex16, dimension( ) dl, complex16, dimension( ) d,complex16, dimension( ) du, complex16, dimension( ) dlf,complex16, dimension( ) df, complex16, dimension( ) duf,complex16, dimension( ) du2, integer, dimension( * ) ipiv,complex16, dimension( ldb, ) b, integer ldb, complex16, dimension(ldx, ) x, integer ldx, double precision rcond, double precision,dimension( * ) ferr, double precision, dimension( * ) berr, complex16,dimension( ) work, double precision, dimension( * ) rwork, integerinfo)

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:

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

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

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

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 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.

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 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

FERR

BERR

WORK

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

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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