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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgttrf (f07cr)

## Purpose

nag_lapack_zgttrf (f07cr) computes the LU $LU$ factorization of a complex n $n$ by n $n$ tridiagonal matrix A $A$.

## Syntax

[dl, d, du, du2, ipiv, info] = f07cr(dl, d, du, 'n', n)
[dl, d, du, du2, ipiv, info] = nag_lapack_zgttrf(dl, d, du, 'n', n)

## Description

nag_lapack_zgttrf (f07cr) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A $A$ as
 A = PLU , $A=PLU ,$
where P $P$ is a permutation matrix, L $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and U $U$ is an upper triangular band matrix, with two superdiagonals.

## 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

## Parameters

### Compulsory Input Parameters

1:     dl( : $:$) – complex array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the matrix A$A$.
2:     d( : $:$) – complex array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the matrix A$A$.
3:     du( : $:$) – complex array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ superdiagonal elements of the matrix A$A$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array d.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

None.

### Output Parameters

1:     dl( : $:$) – complex array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Stores the (n1)$\left(n-1\right)$ multipliers that define the matrix L$L$ of the LU$LU$ factorization of A$A$.
2:     d( : $:$) – complex array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Stores the n$n$ diagonal elements of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
3:     du( : $:$) – complex array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Stores the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
4:     du2(n2${\mathbf{n}}-2$) – complex array
Contains the (n2)$\left(n-2\right)$ elements of the second superdiagonal of U$U$.
5:     ipiv(n) – int64int32nag_int array
Contains the n$n$ pivot indices that define the permutation matrix P$P$. At the i$i$th step, row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$. ipiv(i)${\mathbf{ipiv}}\left(i\right)$ will always be either i$i$ or (i + 1)$\left(i+1\right)$, ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
6:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: dl, 3: d, 4: du, 5: du2, 6: ipiv, 7: info.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, U(i,i)$U\left(i,i\right)$ is exactly zero. The factorization has been completed, but the factor U$U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## Accuracy

The computed factorization satisfies an equation of the form
 A + E = PLU , $A+E=PLU ,$
where
 ‖E‖∞ = O(ε)‖A‖∞ $‖E‖∞=O(ε)‖A‖∞$
and ε $\epsilon$ is the machine precision.
Following the use of this function, nag_lapack_zgttrs (f07cs) can be used to solve systems of equations AX = B $AX=B$ or ATX = B ${A}^{\mathrm{T}}X=B$ or AHX = B ${A}^{\mathrm{H}}X=B$, and nag_lapack_zgtcon (f07cu) can be used to estimate the condition number of A $A$.

The total number of floating point operations required to factorize the matrix A $A$ is proportional to n $n$.
The real analogue of this function is nag_lapack_dgttrf (f07cd).

## Example

```function nag_lapack_zgttrf_example
dl = [ 1 - 2i;
1 + 1i;
2 - 3i;
1 + 1i];
d = [ -1.3 + 1.3i;
-1.3 + 1.3i;
-1.3 + 3.3i;
-0.3 + 4.3i;
-3.3 + 1.3i];
du = [ 2 - 1i;
2 + 1i;
-1 + 1i;
1 - 1i];
[dlOut, dOut, duOut, du2, ipiv, info] = nag_lapack_zgttrf(dl, d, du)
```
```

dlOut =

-0.7800 - 0.2600i
0.1620 - 0.4860i
-0.0452 - 0.0010i
-0.3979 - 0.0562i

dOut =

1.0000 - 2.0000i
1.0000 + 1.0000i
2.0000 - 3.0000i
1.0000 + 1.0000i
-1.3399 + 0.2875i

duOut =

-1.3000 + 1.3000i
-1.3000 + 3.3000i
-0.3000 + 4.3000i
-3.3000 + 1.3000i

du2 =

2.0000 + 1.0000i
-1.0000 + 1.0000i
1.0000 - 1.0000i

ipiv =

2
3
4
5
5

info =

0

```
```function f07cr_example
dl = [ 1 - 2i;
1 + 1i;
2 - 3i;
1 + 1i];
d = [ -1.3 + 1.3i;
-1.3 + 1.3i;
-1.3 + 3.3i;
-0.3 + 4.3i;
-3.3 + 1.3i];
du = [ 2 - 1i;
2 + 1i;
-1 + 1i;
1 - 1i];
[dlOut, dOut, duOut, du2, ipiv, info] = f07cr(dl, d, du)
```
```

dlOut =

-0.7800 - 0.2600i
0.1620 - 0.4860i
-0.0452 - 0.0010i
-0.3979 - 0.0562i

dOut =

1.0000 - 2.0000i
1.0000 + 1.0000i
2.0000 - 3.0000i
1.0000 + 1.0000i
-1.3399 + 0.2875i

duOut =

-1.3000 + 1.3000i
-1.3000 + 3.3000i
-0.3000 + 4.3000i
-3.3000 + 1.3000i

du2 =

2.0000 + 1.0000i
-1.0000 + 1.0000i
1.0000 - 1.0000i

ipiv =

2
3
4
5
5

info =

0

```