Functions
void	magma_print_environment ()
	Print MAGMA version, CUDA version, LAPACK/BLAS library version, available GPU devices, number of threads, date, etc. More...

template<typename FloatT >
void	magma_generate_matrix (magma_opts &opts, Matrix< FloatT > &A, Vector< typename blas::traits< FloatT >::real_t > &sigma)
	Generate an m-by-n test matrix A. More...

Detailed Description

Function Documentation

void magma_print_environment ( )

Print MAGMA version, CUDA version, LAPACK/BLAS library version, available GPU devices, number of threads, date, etc.

Used in testing.

template<typename FloatT >

void magma_generate_matrix	(	magma_opts &	opts,
		Matrix< FloatT > &	A,
		Vector< typename blas::traits< FloatT >::real_t > &	sigma
	)

Generate an m-by-n test matrix A.

Similar to but does not use LAPACK's libtmg.

Parameters

[in]	opts	MAGMA options. Uses matrix, cond, condD; see further details.
[out]	A	Complex array, dimension (lda, n). On output, the m-by-n test matrix A in an lda-by-n array.
[in,out]	sigma	Real array, dimension (min(m,n)) For matrix with "_specified", on input contains user-specified singular or eigenvalues. On output, contains singular or eigenvalues, if known, else set to NaN. sigma is not necesarily sorted.

Further Details

The --matrix command line option specifies the matrix name according to the tables below. Where indicated, names take an optional distribution suffix (%) and an optional scaling suffix (^). The default distribution is rand.

The --cond and --condD command line options specify condition numbers as described below. By default, cond = sqrt( 1/eps ) = 6.7e7 for double. condD is for specialized eigenvalue and SVD tests; by default, condD = 1.

Examples:

./testing_zgemm --matrix rand_small
./testing_zgemm --matrix svd_arith --cond 1e6
./testing_zgemm --matrix svd_logrand_ufl --cond 1e6

In descriptions below:

\(\Sigma\) is a diagonal matrix with entries \(\sigma_i\), for \(i = 1, ..., n\);
\(\Lambda\) is a diagonal matrix with entries \( \lambda_i = \pm \sigma_i \) with random sign, for \(i = 1, ..., n\);
\(U\) and \(V\) are random orthogonal matrices from the Haar distribution (See: Stewart, The efficient generation of random orthogonal matrices with an application to condition estimators, 1980);
\(X\) is a random matrix.

See LAPACK Working Note (LAWN) 41:

Table 5: Test matrices for the nonsymmetric eigenvalue problem
Table 10: Test matrices for the symmetric eigenvalue problem
Table 11: Test matrices for the singular value decomposition

Matrix types

Matrix	Description
`zero`	all entries are 0
`ones`	all entries are 1
`identity`	diagonal entries are 1
`jordan`	diagonal and first subdiagonal entries are 1
`kronecker`	\(A_{ij} = 1 + (m/cond) \delta_{ij} \)
– – –	– – –
`rand^`	matrix entries random uniform on (0, 1) [note 1]
`rands^`	matrix entries random uniform on (-1, 1) [note 1]
`randn^`	matrix entries random normal with mean 0, std 1 [note 1,2]
– – –	– – –
`diag%^`	\(A = \Sigma \)
`svd%^`	\(A = U \Sigma V^H \)
`poev%^`	\(A = V \Sigma V^H \) (eigenvalues positive [note 3], i.e., matrix SPD)
`spd%^`	alias for poev
`heev%^`	\(A = V \Lambda V^H\) (eigenvalues mixed signs)
`syev%^`	alias for heev
`geev%^`	\(A = V T V^H, \) with Schur-form \(T\), \(V\) orthogonal [not yet implemented]
`geevx%^`	\(A = X T X^{-1}, \) with Schur-form \(T\), \(X\) ill-conditioned [not yet implemented]

[1] rand, rands, randn do not use cond, so the condition number is arbitrary.

[2] For randn, \(Expected( \log( cond ) ) = \log( 4.65 n )\) [Edelman, 1988].

[%] optional distribution suffix for Sigma or Lambda

Suffix	Description
`_rand`	\(\sigma_i\) random uniform on (0, 1) [default]
`_rands`	\(\sigma_i\) random uniform on (-1, 1); [note 3]
`_randn`	\(\sigma_i\) random normal with mean 0, std 1; [note 3]
– – –	– – –
`_logrand`	\(\log( \sigma_i )\) uniform on \((\log(1/cond), \log(1))\)
`_arith`	\(\sigma_i = 1 - (i - 1)/(n - 1)(1 - 1/cond); \sigma_{i+1} - \sigma_i \) is constant
`_geo`	\(\sigma_i = (cond)^{ -(i-1)/(n-1) }; \sigma_{i+1} / \sigma_i \) is constant
`_cluster0`	\(\sigma = [ 1, 1/cond, ..., 1/cond ]; \) has 1 unit value, \(n-1\) small values
`_cluster1`	\(\sigma = [ 1, ..., 1, 1/cond ]; \) has \(n-1\) unit values, 1 small value
`_rarith`	`_arith`, reversed order
`_rgeo`	`_geo`, reversed order
`_rcluster0`	`_cluster0`, reversed order
`_rcluster1`	`_cluster1`, reversed order
`_specified`	user specified \(\Sigma\) on input

[3] For _rands, _randn, \(\Sigma\) contains negative values.

[^] optional scaling & modifier suffix

Suffix	Description
`_ufl`	scale near underflow = 1e-308 for double
`_ofl`	scale near overflow = 2e+308 for double
`_small`	scale near sqrt( underflow ) = 1e-154 for double
`_large`	scale near sqrt( overflow ) = 6e+153 for double
`_dominant`	diagonally dominant

For _dominant, set diagonal entries to row or column sum:

\[ A_{ii} = \max( \sum_j | A_{ij} |, \sum_k | A_{ki} | ). \]

Note _dominant changes the singular or eigenvalues, and cond.

condD scaling

Here, scaling by \(D\) is implemented, scaling by \(K\) is not yet implemented. If condD != 1, then: For SVD,

\[ A_0 = U \Sigma V^H, \\ A = A_0 K D, \]

where \(K\) is diagonal such that columns of \(U \Sigma V^H K\) have unit norm, hence \(A^T A\) has unit diagonal, and \(D\) has log-random entries in \(( \log(1/condD), \log(1) )\).

For heev,

\[ A_0 = U \Lambda U^H, \\ A = D K A_0 K D, \]

where \(K\) is diagonal such that \(K A_0 K\) has unit diagonal, and \(D\) is as above.

Note using condD changes the singular or eigenvalues; on output, \(\Sigma\) contains the singular or eigenvalues of \(A_0\), not of \(A\).

See: Demmel and Veselic, Jacobi's method is more accurate than QR, 1992.