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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR: Element $〈value〉$ of the diagonal 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

nag_zgttrf (f07crc) 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 x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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 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 Document

nag_zgttrf (f07crc)

▸▿ 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

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