NAG Library Routine Document

F07JEF (DPTTRS)

is a diagonal matrix, with positive diagonal elements. F07JEF (DPTTRS) 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 Parameters

1: N – INTEGERInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $N \geq 0$ .
2: NRHS – INTEGERInput: On entry: $r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .
Constraint: $NRHS \geq 0$ .
3: D( $*$ ) – REAL (KIND=nag_wp) arrayInput: Note: the dimension 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 $A$ .
4: E( $*$ ) – REAL (KIND=nag_wp) arrayInput: Note: the dimension 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 $A$ .)
5: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array B must be at least $\max (1, NRHS)$ .
On entry: the $n$ by $r$ matrix of right-hand sides $B$ .

On exit: the $n$ by $r$ solution matrix $X$ .
6: LDB – INTEGERInput: On entry: the first dimension of the array B as declared in the (sub)program from which F07JEF (DPTTRS) is called.
Constraint: $LDB \geq \max (1, N)$ .
7: INFO – INTEGEROutput: On exit: $INFO = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

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 routine F07JGF (DPTCON) can be used to estimate the condition number of

A

and F07JHF (DPTRFS) can be used to obtain approximate error bounds.

8 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 routine is F07JSF (ZPTTRS).

9 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 Routine DocumentF07JEF (DPTTRS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07JEF (DPTTRS)