NAG Library Function Document

nag_zgttrf (f07crc)

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: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 0$ .
2: dl[ $\dim$ ] – ComplexInput/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: is overwritten by the $(n - 1)$ multipliers that define the matrix $L$ of the $L U$ factorization of $A$ .
3: d[ $\dim$ ] – ComplexInput/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: is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
4: du[ $\dim$ ] – ComplexInput/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: is overwritten by the $(n - 1)$ elements of the first superdiagonal of $U$ .
5: du2[ $n - 2$ ] – ComplexOutput: On exit: contains the $(n - 2)$ elements of the second superdiagonal of $U$ .
6: ipiv[n] – IntegerOutput: On exit: contains the $n$ pivot indices that define the permutation matrix $P$ . At the $i$ th step, row $i$ of the matrix was interchanged with row $ipiv [i - 1]$ . $ipiv [i - 1]$ will always be either $i$ or $(i + 1)$ , $ipiv [i - 1] = i$ indicating that a row interchange was not performed.
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
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. 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.

7 Accuracy

The computed factorization satisfies an equation of the form

A + E = P L U,

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision.

Following the use of this function, nag_zgttrs (f07csc) can be used to solve systems of equations

A X = B

A^{T} X = B

A^{H} X = B

, and nag_zgtcon (f07cuc) can be used to estimate the condition number of

A

8 Parallelism and Performance

Not applicable.

9 Further Comments

The total number of floating-point operations required to factorize the matrix

A

is proportional to

n

The real analogue of this function is nag_dgttrf (f07cdc).

10 Example

This example factorizes the tridiagonal matrix

A

given by

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array}) .

NAG Library Function Documentnag_zgttrf (f07crc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References