Householder transform of a vector

Functions
float16_t	arm_householder_f16 (const float16_t *pSrc, const float16_t threshold, uint32_t blockSize, float16_t *pOut)
	Householder transform of a half floating point vector. More...
float32_t	arm_householder_f32 (const float32_t *pSrc, const float32_t threshold, uint32_t blockSize, float32_t *pOut)
	Householder transform of a floating point vector. More...
float64_t	arm_householder_f64 (const float64_t *pSrc, const float64_t threshold, uint32_t blockSize, float64_t *pOut)
	Householder transform of a double floating point vector. More...

Description

Computes the Householder transform of a vector x.

The Householder transform of x is a vector v with

\[ v_0 = 1 \]

and a scalar \(\beta\) such that:

\[ P = I - \beta v v^T \]

is an orthogonal matrix and

\[ P x = ||x||_2 e_1 \]

So P is an hyperplane reflection such that the image of x is proportional to \(e_1\).

\(e_1\) is the vector of coordinates:

\[ \begin{pmatrix} 1 \\ 0 \\ \vdots \\ \end{pmatrix} \]

If x is already proportional to \(e_1\) then the matrix P should be the identity.

Thus, \(\beta\) should be 0 and in this case the vector v can also be null.

But how do we detect that x is already proportional to \(e_1\).

If x

\[ x = \begin{pmatrix} x_0 \\ xr \\ \end{pmatrix} \]

where \(xr\) is a vector.

The algorithm is computing the norm squared of this vector:

\[ ||xr||^2 \]

and this value is compared to a threshold. If the value is smaller than the threshold, the algorithm is returning 0 for \(\beta\) and the householder vector.

This threshold is an argument of the function.

Default values are provided in the header dsp/matrix_functions.h like for instance DEFAULT_HOUSEHOLDER_THRESHOLD_F32

Function Documentation

arm_householder_f16()

float16_t arm_householder_f16	(	const float16_t *	pSrc,
		const float16_t	threshold,
		uint32_t	blockSize,
		float16_t *	pOut
	)

Parameters

[in]	pSrc	points to the input vector.
[in]	threshold	norm2 threshold.
[in]	blockSize	dimension of the vector space.
[out]	pOut	points to the output vector.

Returns

beta return the scaling factor beta

arm_householder_f32()

float32_t arm_householder_f32	(	const float32_t *	pSrc,
		const float32_t	threshold,
		uint32_t	blockSize,
		float32_t *	pOut
	)

Parameters

[in]	pSrc	points to the input vector.
[in]	threshold	norm2 threshold.
[in]	blockSize	dimension of the vector space.
[out]	pOut	points to the output vector.

Returns

beta return the scaling factor beta

arm_householder_f64()

float64_t arm_householder_f64	(	const float64_t *	pSrc,
		const float64_t	threshold,
		uint32_t	blockSize,
		float64_t *	pOut
	)

Parameters

[in]	pSrc	points to the input vector.
[in]	threshold	norm2 threshold.
[in]	blockSize	dimension of the vector space.
[out]	pOut	points to the output vector.

Returns

beta return the scaling factor beta