README.md

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# dgttrf

> Compute an `LU` factorization of a real tri diagonal matrix `A` using elimination with partial pivoting and row interchanges.

<section class="usage">

## Usage

```javascript
var dgttrf = require( '@stdlib/lapack/base/dgttrf' );
```

#### dgttrf( N, DL, D, DU, DU2, IPIV )

Computes an `LU` factorization of a real tri diagonal matrix `A` using elimination with partial pivoting and row interchanges.

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var Int32Array = require( '@stdlib/array/int32' );

var DL = new Float64Array( [ 1.0, 1.0 ] );
var D = new Float64Array( [ 2.0, 3.0, 1.0 ] );
var DU = new Float64Array( [ 1.0, 1.0 ] );
var DU2 = new Float64Array( 1 );
var IPIV = new Int32Array( 3 );

dgttrf( 3, DL, D, DU, DU2, IPIV );
// DL => <Float64Array>[ 0.5, 0.4 ]
// D => <Float64Array>[ 2, 2.5, 0.6 ]
// DU => <Float64Array>[ 1, 1 ]
// DU2 => <Float64Array>[ 0 ]
// IPIV => <Int32Array>[ 0, 1, 2 ]
```

The function has the following parameters:

-   **N**: order of matrix `A`.
-   **DL**: the sub diagonal elements of `A` as a [`Float64Array`][mdn-float64array]. On exit, DL is overwritten by the multipliers that define the matrix `L` from the `LU` factorization of `A`.
-   **D**: the diagonal elements of `A` as a [`Float64Array`][mdn-float64array]. On exit, D is overwritten by the diagonal elements of the upper triangular matrix `U` from the `LU` factorization of `A`.
-   **DU**: the super diagonal elements of `A` as a [`Float64Array`][mdn-float64array]. On exit, DU is overwritten by the elements of the first super-diagonal of `U`.
-   **DU2**: On exit, DU2 is overwritten by the elements of the second super-diagonal of `U` as a [`Float64Array`][mdn-float64array].
-   **IPIV**: vector of pivot indices as a [`Int32Array`][mdn-int32array].

Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.

<!-- eslint-disable stdlib/capitalized-comments -->

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var Int32Array = require( '@stdlib/array/int32' );

// Initial arrays...
var DL0 = new Float64Array( [ 0.0, 1.0, 1.0 ] );
var D0 = new Float64Array( [ 0.0, 2.0, 3.0, 1.0 ] );
var DU0 = new Float64Array( [ 0.0, 1.0, 1.0 ] );
var DU20 = new Float64Array( 2 );
var IPIV0 = new Int32Array( 4 );

// Create offset views...
var DL = new Float64Array( DL0.buffer, DL0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var D = new Float64Array( D0.buffer, D0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var DU = new Float64Array( DU0.buffer, DU0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var DU2 = new Float64Array( DU20.buffer, DU20.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var IPIV = new Int32Array( IPIV0.buffer, IPIV0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

dgttrf( 3, DL, D, DU, DU2, IPIV );
// DL0 => <Float64Array>[ 0, 0.5, 0.4 ]
// D0 => <Float64Array>[ 0, 2, 2.5, 0.6 ]
// DU0 => <Float64Array>[ 0, 1, 1 ]
// DU20 => <Float64Array>[ 0, 0 ]
// IPIV0 => <Int32Array>[ 0, 0, 1, 2 ]
```

<!-- lint disable maximum-heading-length -->

#### dgttrf.ndarray( N, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi )

Computes an `LU` factorization of a real tri diagonal matrix `A` using elimination with partial pivoting and row interchanges and alternative indexing semantics.

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var Int32Array = require( '@stdlib/array/int32' );
var dgttrf = require( '@stdlib/lapack/base/dgttrf' );

var DL = new Float64Array( [ 1.0, 1.0 ] );
var D = new Float64Array( [ 2.0, 3.0, 1.0 ] );
var DU = new Float64Array( [ 1.0, 1.0 ] );
var DU2 = new Float64Array( 1 );
var IPIV = new Int32Array( 3 );

dgttrf.ndarray( 3, DL, 1, 0, D, 1, 0, DU, 1, 0, DU2, 1, 0, IPIV, 1, 0 );
// DL => <Float64Array>[ 0.5, 0.4 ]
// D => <Float64Array>[ 2, 2.5, 0.6 ]
// DU => <Float64Array>[ 1, 1 ]
// DU2 => <Float64Array>[ 0 ]
// IPIV => <Int32Array>[ 0, 1, 2 ]
```

The function has the following additional parameters:

-   **sdl**: stride length for `DL`.
-   **odl**:  starting index for `DL`.
-   **sd**: stride length for `D`.
-   **od**:  starting index for `D`.
-   **sdu**: stride length for `DU`.
-   **odu**:  starting index for `DU`.
-   **sdu2**: stride length for `DU2`.
-   **odu2**:  starting index for `DU2`.
-   **si**: stride length for `IPIV`.
-   **oi**:  starting index for `IPIV`.

While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example,

<!-- eslint-disable max-len -->

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var Int32Array = require( '@stdlib/array/int32' );

var DL = new Float64Array( [ 0.0, 1.0, 1.0 ] );
var D = new Float64Array( [ 0.0, 2.0, 3.0, 1.0 ] );
var DU = new Float64Array( [ 0.0, 1.0, 1.0 ] );
var DU2 = new Float64Array( 2 );
var IPIV = new Int32Array( 4 );

dgttrf.ndarray( 3, DL, 1, 1, D, 1, 1, DU, 1, 1, DU2, 1, 1, IPIV, 1, 1 );
// DL => <Float64Array>[ 0, 0.5, 0.4 ]
// D => <Float64Array>[ 0, 2, 2.5, 0.6 ]
// DU => <Float64Array>[ 0, 1, 1 ]
// DU2 => <Float64Array>[ 0, 0 ]
// IPIV => <Int32Array>[ 0, 0, 1, 2 ]
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   Both functions mutate the input arrays `DL`, `D`, `DU`, `DU2` and `IPIV`.

-   Both functions return a status code indicating success or failure. A status code indicates the following conditions:

    -   `0`: factorization was successful.
    -   `<0`: the k-th argument had an illegal value, where `-k` equals the status code value.
    -   `>0`: `U( k, k )` 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, where `k` equals the status code value.

-   `dgttrf()` corresponds to the [LAPACK][LAPACK] routine [`dgttrf`][lapack-dgttrf].

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var Int32Array = require( '@stdlib/array/int32' );
var Float64Array = require( '@stdlib/array/float64' );
var dgttrf = require( '@stdlib/lapack/base/dgttrf' );

var N = 9;

var DL = new Float64Array( [ 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0 ] );

var D = new Float64Array( [ 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 ] );

var DU = new Float64Array( [ 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0 ] );

var DU2 = new Float64Array( N-2 );

var IPIV = new Int32Array( N );

// Perform the `A = LU` factorization:
var info = dgttrf( N, DL, D, DU, DU2, IPIV );

console.log( DL );
console.log( D );
console.log( DU );
console.log( DU2 );
console.log( IPIV );
console.log( info );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
TODO
```

#### TODO

TODO.

```c
TODO
```

TODO

```c
TODO
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
TODO
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[lapack]: https://www.netlib.org/lapack/explore-html/

[lapack-dgttrf]: https://www.netlib.org/lapack/explore-html/d6/d46/group__gttrf_ga8d1e46216e6c861c89bd4328b8be52a1.html#ga8d1e46216e6c861c89bd4328b8be52a1

[mdn-float64array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float64Array

[mdn-int32array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Int32Array

[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray

</section>

<!-- /.links -->