NAG Library Function Document

nag_dpttrs (f07jec)

is a diagonal matrix, with positive diagonal elements. nag_dpttrs (f07jec) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form

U^{T} D U

, where

U

is a unit upper bidiagonal matrix.

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 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: d[ $\dim$ ] – const doubleInput

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 diagonal matrix

D

from the

L D L^{T}

factorization of

A

5: e[ $\dim$ ] – const doubleInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

subdiagonal elements of the unit lower bidiagonal matrix

L

. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix

U

from the

U^{T} D U

factorization of

A

6: 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

matrix of right-hand sides

B

On exit: the

n

r

solution matrix

X

7: 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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

8: 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.

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.

Following the use of this function nag_dptcon (f07jgc) can be used to estimate the condition number of

A

and nag_dptrfs (f07jhc) can be used to obtain approximate error bounds.

8 Parallelism and Performance

nag_dpttrs (f07jec) is not threaded by NAG in any implementation.

nag_dpttrs (f07jec) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

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_zpttrs (f07jsc).

10 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{array}{r} 4.0 & - 2.0 & 0 & 0 & 0 \\ - 2.0 & 10.0 & - 6.0 & 0 & 0 \\ 0 & - 6.0 & 29.0 & 15.0 & 0 \\ 0 & 0 & 15.0 & 25.0 & 8.0 \\ 0 & 0 & 0 & 8.0 & 5.0 \end{array}) and B = (\begin{array}{r} 6.0 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{array}) .

NAG Library Function Documentnag_dpttrs (f07jec)

+− 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_dpttrs (f07jec)