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README.md

dlarf1f

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.

Usage

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.

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.
  • 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).
  • 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 = 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,

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 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 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.

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.
  • 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 = 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 ]

Notes

Examples

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 ) );

C APIs

Usage

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Examples

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