npm package: @stdlib/lapack-base-dgttrf
Description: Compute an `LU` factorization of a real tridiagonal matrix `A` using elimination with partial pivoting and row interchanges
Author: The Stdlib Authors
Latest version: 0.1.1

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# dgttrf

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

<section class="intro">

The `dgttrf` routine computes an LU factorization of a real N-by-N tridiagonal matrix `A` using elimination with partial pivoting and row interchanges. The factorization has the form:

<!-- <equation class="equation" label="eq:lu_decomposition" align="center" raw="A = L U" alt="Equation for LU factorization."> -->

<!-- </equation> -->

where `L` is a product of permutation and unit lower bidiagonal matrices and `U` is upper triangular with non-zeros in only the main diagonal and first two superdiagonals.

For a 5-by-5 tridiagonal matrix `A`, elements are stored in three arrays:

<!-- <equation class="equation" label="eq:matrix_a" align="center" raw="A = \left[\begin{array}{rrrrr}d_1     & du_1     & 0       & 0          & 0 \\dl_1    & d_2      & du_2    & 0          & 0 \\0       & dl_2     & d_3     & du_3       & 0 \\0       & 0        & dl_3    & d_4        & du_4 \\0       & 0        & 0       & dl_4       & d_5\end{array}\right]" alt="Representation of matrix A."> -->

<!-- </equation> -->

where:

-   `dl` contains the subdiagonal elements.
-   `d` contains the diagonal elements.
-   `du` contains the superdiagonal elements.

After factorization, the elements of `L` and `U` are used to update the input arrays, where:

-   `dl` contains the multipliers that define unit lower bidiagonal matrix `L`.
-   `d` contains the diagonal elements of `U`.
-   `du` and `du2` contain the elements of `U` on the first and second superdiagonals.

The resulting `L` and `U` matrices have the following structure:

<!-- <equation class="equation" label="eq:matrix_l" align="center" raw="L = \left[\begin{array}{rrrrr}1      & 0      & 0      & 0      & 0 \\l_1    & 1      & 0      & 0      & 0 \\0      & l_2    & 1      & 0      & 0 \\0      & 0      & l_3    & 1      & 0 \\0      & 0      & 0      & l_4    & 1\end{array}\right]" alt="Representation of matrix L as derived from DL."> -->

<!-- </equation> -->

<!-- <equation class="equation" label="eq:matrix_u" align="center" raw="U = \left[\begin{array}{rrrrr}u_{1,1} & u_{1,2} & u_{1,3} & 0       & 0 \\0       & u_{2,2} & u_{2,3} & u_{2,4} & 0 \\0       & 0       & u_{3,3} & u_{3,4} & u_{3,5} \\0       & 0       & 0       & u_{4,4} & u_{4,5} \\0       & 0       & 0       &  0      & u_{5,5}\end{array}\right]" alt="Representation of matrix U as derived from D, DU, DU2."> -->

<!-- </equation> -->

where the `l(i)` values are stored in `DL`, the diagonal elements `u(i,i)` are stored in `D`, and the superdiagonal elements `u(i,i+1)` and `u(i,i+2)` are stored in `DU` and `DU2`, respectively.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/lapack-base-dgttrf
```

</section>

<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 tridiagonal matrix `A` using elimination with partial pivoting and row interchanges.

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

var DL = new Float64Array( [ 6.0, 6.0 ] );
var D = new Float64Array( [ 20.0, 30.0, 10.0 ] );
var DU = new Float64Array( [ 8.0, 8.0 ] );
var DU2 = new Float64Array( [ 0.0 ] );
var IPIV = new Int32Array( [ 0, 0, 0 ] );

/*
    A = [
        [ 20.0, 8.0,  0.0  ],
        [ 6.0,  30.0, 8.0  ],
        [ 0.0,  6.0,  10.0 ]
    ]
*/

dgttrf( 3, DL, D, DU, DU2, IPIV );
// DL => <Float64Array>[ 0.3, ~0.22 ]
// D => <Float64Array>[ 20.0, 27.6, ~8.26 ]
// DU => <Float64Array>[ 8.0, 8.0 ]
// DU2 => <Float64Array>[ 0.0 ]
// IPIV => <Int32Array>[ 0, 1, 2 ]
```

The function has the following parameters:

-   **N**: number of rows/columns in `A`.
-   **DL**: the first sub diagonal of `A` as a [`Float64Array`][mdn-float64array]. Should have `N-1` indexed elements. `DL` is overwritten by the multipliers that define the matrix `L` from the `LU` factorization of `A`.
-   **D**: the diagonal of `A` as a [`Float64Array`][mdn-float64array]. Should have `N` indexed elements. `D` is overwritten by the diagonal elements of the upper triangular matrix `U` from the `LU` factorization of `A`.
-   **DU**: the first super-diagonal of `A` as a [`Float64Array`][mdn-float64array]. Should have `N-1` indexed elements. `DU` is overwritten by the elements of the first super-diagonal of `U`.
-   **DU2**: the second super-diagonal of `U`  as a [`Float64Array`][mdn-float64array]. Should have `N-2` indexed elements. `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]. Should have `N` indexed elements.

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

// Initial arrays...
var DL0 = new Float64Array( [ 0.0, 6.0, 6.0 ] );
var D0 = new Float64Array( [ 0.0, 20.0, 30.0, 10.0 ] );
var DU0 = new Float64Array( [ 0.0, 8.0, 8.0 ] );
var DU20 = new Float64Array( [ 0.0, 0.0 ] );
var IPIV0 = new Int32Array( [ 0, 0, 0, 0 ] );

/*
    A = [
        [ 20.0, 8.0,  0.0  ],
        [ 6.0,  30.0, 8.0  ],
        [ 0.0,  6.0,  10.0 ]
    ]
*/

// 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, 0.3, ~0.22 ]
// D0 => <Float64Array>[ 0.0, 20.0, 27.6, ~8.26 ]
// DU0 => <Float64Array>[ 0.0, 8.0, 8.0 ]
// DU20 => <Float64Array>[ 0.0, 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 tridiagonal 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( [ 6.0, 6.0 ] );
var D = new Float64Array( [ 20.0, 30.0, 10.0 ] );
var DU = new Float64Array( [ 8.0, 8.0 ] );
var DU2 = new Float64Array( [ 0.0 ] );
var IPIV = new Int32Array( [ 0, 0, 0 ] );

/*
    A = [
        [ 20.0, 8.0,  0.0  ],
        [ 6.0,  30.0, 8.0  ],
        [ 0.0,  6.0,  10.0 ]
    ]
*/

dgttrf.ndarray( 3, DL, 1, 0, D, 1, 0, DU, 1, 0, DU2, 1, 0, IPIV, 1, 0 );
// DL => <Float64Array>[ 0.3, ~0.22 ]
// D => <Float64Array>[ 20.0, 27.6, ~8.26 ]
// DU => <Float64Array>[ 8.0, 8.0 ]
// DU2 => <Float64Array>[ 0.0 ]
// 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 dgttrf = require( '@stdlib/lapack-base-dgttrf' );

var DL = new Float64Array( [ 0.0, 6.0, 6.0 ] );
var D = new Float64Array( [ 0.0, 20.0, 30.0, 10.0 ] );
var DU = new Float64Array( [ 0.0, 8.0, 8.0 ] );
var DU2 = new Float64Array( [ 0.0, 0.0 ] );
var IPIV = new Int32Array( [ 0, 0, 0, 0 ] );

/*
    A = [
        [ 20.0, 8.0,  0.0  ],
        [ 6.0,  30.0, 8.0  ],
        [ 0.0,  6.0,  10.0 ]
    ]
*/

dgttrf.ndarray( 3, DL, 1, 1, D, 1, 1, DU, 1, 1, DU2, 1, 1, IPIV, 1, 1 );
// DL => <Float64Array>[ 0.0, 0.3, ~0.22 ]
// D => <Float64Array>[ 0.0, 20.0, 27.6, ~8.26 ]
// DU => <Float64Array>[ 0.0, 8.0, 8.0 ]
// DU2 => <Float64Array>[ 0.0, 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. The status code indicates the following conditions:

    -   `0`: factorization was successful.
    -   `>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 );

/*
    A = [
        [ 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
        [ 3.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
        [ 0.0, 3.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
        [ 0.0, 0.0, 3.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0 ],
        [ 0.0, 0.0, 0.0, 3.0, 1.0, 4.0, 0.0, 0.0, 0.0 ],
        [ 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 4.0, 0.0, 0.0 ],
        [ 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 4.0, 0.0 ],
        [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 4.0 ],
        [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0 ],
    ]
*/

// 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 -->

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* * *

## Notice

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For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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

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