# NAG C Library Function Document

## 1Purpose

nag_dgttrf (f07cdc) computes the $LU$ factorization of a real $n$ by $n$ tridiagonal matrix $A$.

## 2Specification

 #include #include
 void nag_dgttrf (Integer n, double dl[], double d[], double du[], double du2[], Integer ipiv[], NagError *fail)

## 3Description

nag_dgttrf (f07cdc) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals.

## 4References

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

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{dl}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
On exit: is overwritten by the $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
3:    $\mathbf{d}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 $LU$ factorization of $A$.
4:    $\mathbf{du}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$.
On exit: is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
5:    $\mathbf{du2}\left[{\mathbf{n}}-2\right]$doubleOutput
On exit: contains the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
6:    $\mathbf{ipiv}\left[{\mathbf{n}}\right]$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 ${\mathbf{ipiv}}\left[i-1\right]$. ${\mathbf{ipiv}}\left[i-1\right]$ will always be either $i$ or $\left(i+1\right)$, ${\mathbf{ipiv}}\left[i-1\right]=i$ indicating that a row interchange was not performed.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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 $〈\mathit{\text{value}}〉$ 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.

## 7Accuracy

The computed factorization satisfies an equation of the form
 $A+E=PLU ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision.
Following the use of this function, nag_dgttrs (f07cec) can be used to solve systems of equations $AX=B$ or ${A}^{\mathrm{T}}X=B$, and nag_dgtcon (f07cgc) can be used to estimate the condition number of $A$.

## 8Parallelism and Performance

nag_dgttrf (f07cdc) is not threaded in any implementation.

The total number of floating-point operations required to factorize the matrix $A$ is proportional to $n$.
The complex analogue of this function is nag_zgttrf (f07crc).

## 10Example

This example factorizes the tridiagonal matrix $A$ given by
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 .$

### 10.1Program Text

Program Text (f07cdce.c)

### 10.2Program Data

Program Data (f07cdce.d)

### 10.3Program Results

Program Results (f07cdce.r)