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dlarf

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

A Householder transformation (or an elementary reflector) is a linear transformation that describes a reflection about a plane or a hyperplane containing the origin. This routine applies an elementary reflector of the form

$$H = I - \tau v v^{\mathsf{T}}$$

Depending on side, C is overwritten by either H * C or C * H.

Installation

npm install @stdlib/lapack-base-dlarf

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

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

dlarf( 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.

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 = dlarf( 'row-major', 'left', 4, 3, V, 1, 1.0, C, 3, work );
// returns <Float64Array>[ -1.5, -1.5, -1.5, -0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 1.5, 1.5, 1.5 ]

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.
• 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.
• LDC: stride of the first dimension of C (a.k.a., leading dimension of the matrix C). Must be at least max(1,N) when order is 'row-major' and at least max(1,M) when order is 'column-major'.
• work: workspace 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,

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 = dlarf( 'row-major', 'left', 4, 3, V, -1, 1.0, C, 3, work );
// returns <Float64Array>[ 0.2, ~3.08, ~5.96, 0.8, ~3.12, ~5.44, 1.4, ~3.16, ~4.92, 2.0, 3.2, 4.4 ]

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

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 = dlarf( 'row-major', 'left', 4, 3, V, 2, 1.0, C, 3, work );
// returns <Float64Array>[ -1.5, -1.5, -1.5, -0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 1.5, 1.5, 1.5 ]

Note that indexing is relative to the first index. To introduce an offset, use typed array views.

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 out = dlarf( 'row-major', 'left', 4, 3, V1, 1, 1.0, C1, 3, work1 );
// C0 => <Float64Array>[ 0.0, -1.5, -1.5, -1.5, -0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 1.5, 1.5, 1.5 ]

dlarf.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.

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 = dlarf.ndarray( 'left', 4, 3, V, 1, 0, 1.0, C, 3, 1, 0, work, 1, 0 );
// returns <Float64Array>[ -1.5, -1.5, -1.5, -0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 1.5, 1.5, 1.5 ]

The function has the following 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.
• sv: stride length for V.
• ov: starting index for V.
• tau: scalar constant.
• C: input matrix as a 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.
• 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 views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example,

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 = dlarf.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, -1.5, -1.5, -1.5, -0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 1.5, 1.5, 1.5 ]

Notes

• dlarf() corresponds to the LAPACK function dlarf.

Examples

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

// 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:
dlarf( order, 'left', shape[ 0 ], shape[ 1 ], V, 1, 1.0, C, strides[ 0 ], work );
console.log( ndarray2array( C, shape, strides, 0, order ) );

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C APIs

Usage

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Examples

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

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