npm package: @stdlib/lapack-base-dlarf1f
Description: LAPACK routine to apply a real elementary reflector `H = I - tau * v * v^T` to a real M by N matrix `C`.
Author: The Stdlib Authors
Latest version: 0.1.1

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# dlarf1f

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> Apply a real elementary reflector `H = I - tau * v * v^T` to a real M by N matrix `C`.

<section class="intro">

A **Householder transformation** (or an **elementary reflector**) is a linear transformation that describes a reflection about a plane or a hyperplane containing the origin.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/lapack-base-dlarf1f
```

</section>

<section class="usage">

## Usage

```javascript
var dlarf1f = require( '@stdlib/lapack-base-dlarf1f' );
```

#### dlarf1f( order, side, M, N, V, strideV, tau, C, LDC, work )

Applies a real elementary reflector `H = I - tau * v * v^T` to a real M by N matrix `C`.

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

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

var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
var work = new Float64Array( 3 );

var out = dlarf1f( 'row-major', 'left', 4, 3, V, 1, 1.0, C, 3, work );
// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
```

The function has the following parameters:

-   **order**: storage layout.
-   **side**: specifies the side of multiplication with `C`.
-   **M**: number of rows in `C`.
-   **N**: number of columns in `C`.
-   **V**: the vector `v` as a [`Float64Array`][mdn-float64array].
-   **strideV**: stride length for `V`. If `strideV` is negative, the elements of `V` are accessed in reverse order.
-   **tau**: scalar constant.
-   **C**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
-   **LDC**: stride of the first dimension of `C` (a.k.a., leading dimension of the matrix `C`).
-   **work**: workspace [`Float64Array`][mdn-float64array].

When `side` is `'left'`,

-   `work` should have `N` indexed elements.
-   `V` should have `1 + (M-1) * abs(strideV)` indexed elements.
-   `C` is overwritten by `H * C`.

When `side` is `'right'`,

-   `work` should have `M` indexed elements.
-   `V` should have `1 + (N-1) * abs(strideV)` indexed elements.
-   `C` is overwritten by `C * H`.

The sign of the increment parameter `strideV` determines the order in which elements of `V` are accessed. For example, to access elements in reverse order,

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

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

var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
var V = new Float64Array( [ 0.5, 0.4, 0.3, 0.2 ] );
var work = new Float64Array( 3 );

var out = dlarf1f( 'row-major', 'left', 4, 3, V, -1, 1.0, C, 3, work );
// returns <Float64Array>[ ~-3.80, -8.6, ~-13.4, ~0.56, 1.92, ~3.28, ~1.08, ~1.56, ~2.04, ~1.60, ~1.20, ~0.80 ]
```

To perform strided access over `V`, provide an `abs(strideV)` value greater than one. For example, to access every other element in `V`,

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

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

var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
var V = new Float64Array( [ 0.5, 999, 0.5, 999, 0.5, 999, 0.5 ] );
var work = new Float64Array( 3 );

var out = dlarf1f( 'row-major', 'left', 4, 3, V, 2, 1.0, C, 3, work );
// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
```

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, max-len -->

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

// Initial arrays...
var C0 = new Float64Array( [ 0.0, 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
var V0 = new Float64Array( [ 0.0, 0.5, 0.5, 0.5, 0.5 ] );
var work0 = new Float64Array( [ 0.0, 0.0, 0.0, 0.0 ] );

// Create offset views...
var C1 = new Float64Array( C0.buffer, C0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var V1 = new Float64Array( V0.buffer, V0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var work1 = new Float64Array( work0.buffer, work0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

var our = dlarf1f( 'row-major', 'left', 4, 3, V1, 1, 1.0, C1, 3, work1 );
// C0 => <Float64Array>[ 0.0, -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
```

#### dlarf1f.ndarray( side, M, N, V, sv, ov, tau, C, sc1, sc2, oc, work, sw, ow )

Applies a real elementary reflector `H = I - tau * v * v^T` to a real M by N matrix `C` using alternative indexing semantics.

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

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

var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
var work = new Float64Array( 3 );

var out = dlarf1f.ndarray( 'left', 4, 3, V, 1, 0, 1.0, C, 3, 1, 0, work, 1, 0 );
// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
```

The function has the following additional parameters:

-   **side**: specifies the side of multiplication with `C`.
-   **M**: number of rows in `C`.
-   **N**: number of columns in `C`.
-   **V**: the vector `v` as a [`Float64Array`][mdn-float64array].
-   **sv**: stride length for `V`.
-   **ov**: starting index for `V`.
-   **tau**: scalar constant.
-   **C**: input matrix as a [`Float64Array`][mdn-float64array].
-   **sc1**: stride of the first dimension of `C`.
-   **sc2**: stride of the second dimension of `C`.
-   **oc**: starting index for `C`.
-   **work**: workspace array as a [`Float64Array`][mdn-float64array].
-   **sw**: stride length for `work`.
-   **ow**: starting index for `work`.

When `side` is `'left'`,

-   `work` should have `N` indexed elements.
-   `V` should have `1 + (M-1) * abs(sv)` indexed elements.
-   `C` is overwritten by `H * C`.

When `side` is `'right'`,

-   `work` should have `M` indexed elements.
-   `V` should have `1 + (N-1) * abs(sv)` indexed elements.
-   `C` is overwritten by `C * H`.

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 C = new Float64Array( [ 0.0, 0.0, 0.0, 0.0, 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
var V = new Float64Array( [ 0.0, 0.0, 0.5, 0.5, 0.5, 0.5 ] );
var work = new Float64Array( [ 0.0, 0.0, 0.0, 0.0 ] );

var out = dlarf1f.ndarray( 'left', 4, 3, V, 1, 2, 1.0, C, 3, 1, 4, work, 1, 0 );
// C => <Float64Array>[ 0.0, 0.0, 0.0, 0.0, -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   `dlarf1f()` corresponds to the [LAPACK][LAPACK] function [`dlarf1f`][lapack-dlarf1f].

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

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

```javascript
var Float64Array = require( '@stdlib/array-float64' );
var ndarray2array = require( '@stdlib/ndarray-base-to-array' );
var shape2strides = require( '@stdlib/ndarray-base-shape2strides' );
var dlarf1f = require( '@stdlib/lapack-base-dlarf1f' );

// Specify matrix meta data:
var shape = [ 4, 3 ];
var order = 'row-major';
var strides = shape2strides( shape, order );

// Create a matrix stored in linear memory:
var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
console.log( ndarray2array( C, shape, strides, 0, order ) );

// Define the vector `v` and a workspace array:
var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
var work = new Float64Array( 3 );

// Apply the elementary reflector:
dlarf1f( order, 'left', shape[ 0 ], shape[ 1 ], V, 1, 1.0, C, strides[ 0 ], work );
console.log( ndarray2array( C, shape, strides, 0, order ) );
```

</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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<section class="main-repo" >

* * *

## 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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## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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

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