imate.trlinfrac

imate.trlinfrac(A, gram=False, p=1.0, coeff=[1.0, 0.0, 1.0, 1.0], return_info=False, method='eigenvalue', **options)

Trace of the linear fractional transformation of matrix or linear operator.

Given the matrix or the linear operator \(\mathbf{A}\) and the real exponent \(p\), the following is computed:

\[\mathrm{trace} f \left(\mathbf{A}^p \right),\]

where the function \(f\) is defined by

\[f(\lambda) = \frac{a \lambda + b}{c \lambda +d},\]

provided that \(ad - bc \neq 0\).

If gram is True, then \(\mathbf{A}\) in the above is replaced by the Gramian matrix \(\mathbf{A}^{\intercal} \mathbf{A}\), and the following is instead computed:

\[\mathrm{trace} f \left((\mathbf{A}^{\intercal}\mathbf{A})^p \right).\]

If \(\mathbf{A} = \mathbf{A}(t)\) is a linear operator of the class imate.AffineMatrixFunction with the parameter \(t\), then for an input tuple \(t = (t_1, \dots, t_q)\), an array output of the size \(q\) is returned, namely:

\[\mathrm{trace} f \left((\mathbf{A}(t_i))^p \right), \quad i=1, \dots, q.\]

Parameters:

Anumpy.ndarray, scipy.sparse, imate.Matrix, or imate.AffineMatrixFunction

A sparse or dense matrix or linear operator. The linear operators imate.Matrix and imate.AffineMatrixFunction can be used only if method=slq. See details in slq method. If method=cholesky, the matrix A should be positive-definite. If method=slq and gram=False, the input matrix A should be symmetric. If gram=True, the input matrix can be non-square.

grambool, default=False

If True, the trace-exponential of the Gramian matrix, \((\mathbf{A}^{\intercal}\mathbf{A})^p\), is computed. The Gramian matrix itself is not directly computed. If False, the trace-exponential of \(\mathbf{A}^p\) is computed.

pfloat, default=1.0

The real exponent \(p\) in \(\mathbf{A}^p\).

coeff(tuple, list, numpy.array), default=[1,0, 0.0, 1.0, 1.0]

A list, tuple, or array of four float numbers representing the coefficients \(a\), \(b\), \(c\), and \(d\), respectively.

return_infobool, default=False

If True, this function also returns a dictionary containing information about the inner computation, such as process time, algorithm settings, etc. See the documentation for each method for details.

method{‘eigenvalue’, ‘slq’}, default=’eigenvalue’

The method of computing trace-exponential. See documentation for each method:

• eigenvalue

• slq

options**kwargs

Extra arguments that are specific to each method. See the documentation for each method for details.

Returns:

trlinfracfloat or numpy.array

Trace-exponential of matrix. If method=slq and if A is of type imate.AffineMatrixFunction with an array of parameters, then the output is an array.

infodict

(Only if return_info is True) A dictionary of information with at least the following keys. Further keys specific to each method can be found in the documentation of each method.

• matrix:

  • data_type: str, {float32, float64, float128}, type of the matrix data.

  • gram: bool, whether the matrix A or its Gramian is considered.

  • exponent: float, the exponent p in \(\mathbf{A}^p\).

  • size: int, The size of matrix A.

  • sparse: bool, whether the matrix A is sparse or dense.

  • nnz: int, if A is sparse, the number of non-zero elements of A.

  • density: float, if A is sparse, the density of A, which is the nnz divided by size squared.

  • num_inquiries: int, The size of inquiries of each parameter of the linear operator A. If A is a matrix, this is always 1. For more details see slq method.

• device:

  • num_cpu_threads: int, number of CPU threads used in shared memory parallel processing.

  • num_gpu_devices: int, number of GPU devices used in the multi-GPU (GPU farm) computation.

  • num_gpu_multiprocessors: int, number of GPU multi-processors.

  • num_gpu_threads_per_multiprocessor: int, number of GPU threads on each GPU multi-processor.

• time:

  • tot_wall_time: float, total elapsed time of computation.

  • alg_wall_time: float, elapsed time of computation during only the algorithm execution.

  • cpu_proc_time: float, CPU processing time of computation.

• solver:

  • version: str, version of imate.

  • method: str, method of computation.

Raises:

ImportError

If the package has not been compiled with GPU support, but gpu is True. Either set gpu to False to use the existing installed package. Alternatively, export the environment variable USE_CUDA=1 and recompile the source code of the package.

See also

imate.logdet

imate.trace

imate.trexp

imate.traceinv

imate.schatten

Notes

Method of Computation:

See documentation for each method below.

• eigenvalue: uses spectral decomposition. Suitable for small matrices (\(n < 2^{12}\)). The solution is exact.

• slq: uses stochastic Lanczos quadrature (SLQ), which is a randomized algorithm. Can be used on very large matrices (\(n > 2^{12}\)). The solution is an approximation.

Input Matrix:

The input A can be either of:

• A matrix, such as numpy.ndarray, or scipy.sparse.

• A linear operator representing a matrix using imate.Matrix ( only if method=slq).

• A linear operator representing a one-parameter family of an affine matrix function \(t \mapsto \mathbf{A} + t\mathbf{B}\), using imate.AffineMatrixFunction (only if method=slq).

Output:

The output is a scalar. However, if A is the linear operator of the type imate.AffineMatrixFunction representing the matrix function \(\mathbf{A}(t) = \mathbf{A} + t \mathbf{B}\), then if the parameter \(t\) is given as the tuple \(t = (t_1, \dots, t_q)\), then the output of this function is an array of size \(q\) corresponding to the trace-exponential of each \(\mathbf{A}(t_i)\).

Note

When A represents \(\mathbf{A}(t) = \mathbf{A} + t \mathbf{I}\), where \(\mathbf{I}\) is the identity matrix, and \(t\) is given by a tuple \(t = (t_1, \dots, t_q)\), by setting method=slq, the computational cost of an array output of size q is the same as computing for a single \(t_i\). Namely, the trace-exponential of only \(\mathbf{A}(t_1)\) is computed, and the trace-exponential of the rest of \(q=2, \dots, q\) are obtained from the result of \(t_1\) immediately.

Examples

Sparse matrix:

Compute the trace-exponential of a sample sparse Toeplitz matrix created by imate.toeplitz() function.

>>> # Import packages
>>> from imate import toeplitz, trlinfrac

>>> # Generate a sample matrix (a toeplitz matrix)
>>> A = toeplitz(2, 1, size=100)

>>> # Compute trace-exponential with eigenvalue method (default method)
>>> trlinfrac(A)
138.6294361119891

Alternatively, compute the trace-exponential of \(\mathbf{A}^{\intercal} \mathbf{A}\):

>>> # Compute trace-exponential of the Gramian to the power of 3:
>>> trlinfrac(A, p=3, gram=True)
831.7766166719346

Output information:

Print information about the inner computation:

>>> ld, info = trlinfrac(A, return_info=True)
>>> print(ld)
138.6294361119891

>>> # Print dictionary neatly using pprint
>>> from pprint import pprint
>>> pprint(info)
{
    'matrix': {
        'data_type': b'float64',
        'density': 0.0298,
        'exponent': 1.0,
        'gram': False,
        'nnz': 298,
        'num_inquiries': 1,
        'size': 100,
        'sparse': True
    },
    'solver': {
        'cholmod_used': True,
        'method': 'eigenvalue',
        'version': '0.13.0'
    },
    'device': {
        'num_cpu_threads': 8,
        'num_gpu_devices': 0,
        'num_gpu_multiprocessors': 0,
        'num_gpu_threads_per_multiprocessor': 0
    },
    'time': {
        'alg_wall_time': 0.000903537031263113,
        'cpu_proc_time': 0.0010093420000032438,
        'tot_wall_time': 0.000903537031263113
    }
}

Large matrix:

Compute trace-exponential of a very large sparse matrix using SLQ method. This method does not compute race-exponential exactly, rather, the result is an approximation using Monte-Carlo sampling. The following example uses at least 100 samples.

>>> # Generate a matrix of size one million
>>> A = toeplitz(2, 1, size=1000000, gram=True)

>>> # Approximate trace-exponential using stochastic Lanczos quadrature
>>> # with at least 100 Monte-Carlo sampling
>>> ld, info = trlinfrac(A, method='slq', min_num_samples=100,
...                      max_num_samples=200, return_info=True)
>>> print(ld)
1386187.5751816272

>>> # Find the time it took to compute the above
>>> print(info['time'])
{
    'tot_wall_time': 15.908390650060028,
    'alg_wall_time': 15.890228271484375,
    'cpu_proc_time': 116.93080989600001
}

Matrix operator:

The following example uses an object of imate.Matrix. Note that this can be only applied to method=slq. See further details in slq method.

>>> # Import matrix operator
>>> from imate import toeplitz, trlinfrac, Matrix

>>> # Generate a sample matrix (a toeplitz matrix)
>>> A = toeplitz(2, 1, size=100, gram=True)

>>> # Create a matrix operator object from matrix A
>>> Aop = Matrix(A)

>>> # Compute trace-exponential of Aop
>>> trlinfrac(Aop, method='slq')
141.52929878934194

Affine matrix operator:

The following example uses an object of imate.AffineMatrixFunction to create the linear operator:

\[t \mapsto \mathbf{A} + t \mathbf{I}\]

Note that this can be only applied to method=slq. See further details in slq method.

>>> # Import affine matrix function
>>> from imate import toeplitz, trlinfrac, AffineMatrixFunction

>>> # Generate a sample matrix (a toeplitz matrix)
>>> A = toeplitz(2, 1, size=100, gram=True)

>>> # Create a matrix operator object from matrix A
>>> Aop = AffineMatrixFunction(A)

>>> # A list of parameters t to pass to Aop
>>> t = [-1.0, 0.0, 1.0]

>>> # Compute trace-exp of Aop for all parameters t
>>> trlinfrac(Aop, method='slq', parameters=t)
array([ 68.71411681, 135.88356906, 163.44156683])