NAG Library Function Document

nag_dgtsv (f07cac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_dgtsv (f07cac) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

tridiagonal matrix and

X

and

B

are

n

r

matrices.

2 Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dgtsv (Nag_OrderType order, Integer n, Integer nrhs, double dl[], double d[], double du[], double b[], Integer pdb, NagError *fail)

3 Description

nag_dgtsv (f07cac) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations

A X = B

. The matrix

A

is factorized as

A = P L U

, where

P

is a permutation matrix,

L

is unit lower triangular with at most one nonzero subdiagonal element per column, and

U

is an upper triangular band matrix, with two superdiagonals.

Note that the equations

A^{T} X = B

may be solved by interchanging the order of the arguments du and dl.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

5 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: n – IntegerInput

On entry:

n

, the number of linear equations, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: nrhs – IntegerInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

4: dl[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array dl must be at least

\max (1, n - 1)

On entry: must contain the

(n - 1)

subdiagonal elements of the matrix

A

On exit: if no constrains are violated, dl is overwritten by the (

n - 2

) elements of the second superdiagonal of the upper triangular matrix

U

from the

L U

factorization of

A

, in

dl [0], dl [1], \dots, dl [n - 3]

5: d[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array d must be at least

\max (1, n)

On entry: must contain the

n

diagonal elements of the matrix

A

On exit: if no constraints are violated, d is overwritten by the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

6: du[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array du must be at least

\max (1, n - 1)

On entry: must contain the

(n - 1)

superdiagonal elements of the matrix

A

On exit: if no constraints are violated, du is overwritten by the

(n - 1)

elements of the first superdiagonal of

U

7: b[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

right-hand side matrix

B

On exit: if

fail . code =

NE_NOERROR, the

n

r

solution matrix

X

8: pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $nrhs = ⟨value⟩$ .
Constraint: $nrhs \geq 0$ .
On entry, $pdb = ⟨value⟩$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdb = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
On entry, $pdb = ⟨value⟩$ and $nrhs = ⟨value⟩$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR: $U (⟨value⟩, ⟨value⟩)$ is exactly zero, and the solution has not been computed. The factorization has not been completed unless $n = ⟨value⟩$ .

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

Alternatives to nag_dgtsv (f07cac), which return condition and error estimates are nag_real_tridiag_lin_solve (f04bcc) and nag_dgtsvx (f07cbc).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

The complex analogue of this function is nag_zgtsv (f07cnc).

10 Example

This example solves the equations

A x = b,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) and b = (\begin{matrix} 2.7 \\ - 0.5 \\ 2.6 \\ 0.6 \\ 2.7 \end{matrix}) .

NAG Library Function Documentnag_dgtsv (f07cac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_dgtsv (f07cac)