# sporco.linalg¶

Linear algebra functions

Functions

 complex_dtype(dtype) Construct the corresponding complex dtype for a given real dtype, e.g. pyfftw_byte_aligned(array[, dtype, n]) Construct a byte-aligned array for efficient use by pyfftw. pyfftw_empty_aligned(shape, dtype[, order, n]) Construct an empty byte-aligned array for efficient use by pyfftw. pyfftw_rfftn_empty_aligned(shape, axes, dtype) Construct an empty byte-aligned array for efficient use by pyfftw functions pyfftw.interfaces.numpy_fft.rfftn and pyfftw.interfaces.numpy_fft.irfftn. fftn(a[, s, axes, norm]) Compute the multi-dimensional discrete Fourier transform. ifftn(a[, s, axes, norm]) Compute the multi-dimensional inverse discrete Fourier transform. rfftn(a[, s, axes, norm]) Compute the multi-dimensional discrete Fourier transform for real input. irfftn(a[, s, axes, norm]) Compute the inverse of the multi-dimensional discrete Fourier transform for real input. dctii(x[, axes]) Compute a multi-dimensional DCT-II over specified array axes. idctii(x[, axes]) Compute a multi-dimensional inverse DCT-II over specified array axes. fftconv(a, b[, axes]) Compute a multi-dimensional convolution via the Discrete Fourier Transform. inner(x, y[, axis]) Compute inner product of x and y on specified axis, equivalent to np.sum(x * y, axis=axis, keepdims=True). dot(a, b[, axis]) Compute the matrix product of a and the specified axes of b, with broadcasting over the remaining axes of b. solvedbi_sm(ah, rho, b[, c, axis]) Solve a diagonal block linear system with a scaled identity term using the Sherman-Morrison equation. solvedbi_sm_c(ah, a, rho[, axis]) Compute cached component used by solvedbi_sm. solvedbd_sm(ah, d, b[, c, axis]) Solve a diagonal block linear system with a diagonal term using the Sherman-Morrison equation. solvedbd_sm_c(ah, a, d[, axis]) Compute cached component used by solvedbd_sm. solvemdbi_ism(ah, rho, b, axisM, axisK) Solve a multiple diagonal block linear system with a scaled identity term by iterated application of the Sherman-Morrison equation. solvemdbi_rsm(ah, rho, b, axisK[, dimN]) Solve a multiple diagonal block linear system with a scaled identity term by repeated application of the Sherman-Morrison equation. solvemdbi_cg(ah, rho, b, axisM, axisK[, …]) Solve a multiple diagonal block linear system with a scaled identity term using Conjugate Gradient (CG) via scipy.sparse.linalg.cg. lu_factor(A, rho[, check_finite]) Compute LU factorisation of either $$A^T A + \rho I$$ or $$A A^T + \rho I$$, depending on which matrix is smaller. lu_solve_ATAI(A, rho, b, lu, piv[, check_finite]) Solve the linear system $$(A^T A + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A^T A + \rho I)X = B$$ using scipy.linalg.lu_solve. lu_solve_AATI(A, rho, b, lu, piv[, check_finite]) Solve the linear system $$(A A^T + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A A^T + \rho I)X = B$$ using scipy.linalg.lu_solve. cho_factor(A, rho[, lower, check_finite]) Compute Cholesky factorisation of either $$A^T A + \rho I$$ or $$A A^T + \rho I$$, depending on which matrix is smaller. cho_solve_ATAI(A, rho, b, c, lwr[, check_finite]) Solve the linear system $$(A^T A + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A^T A + \rho I)X = B$$ using scipy.linalg.cho_solve. cho_solve_AATI(A, rho, b, c, lwr[, check_finite]) Solve the linear system $$(A A^T + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A A^T + \rho I)X = B$$ using scipy.linalg.cho_solve. zpad(x, pd, ax) Zero-pad array x with pd = (leading, trailing) zeros on axis ax. Gax(x, ax) Compute gradient of x along axis ax. GTax(x, ax) Compute transpose of gradient of x along axis ax. GradientFilters(ndim, axes, axshp[, dtype]) Construct a set of filters for computing gradients in the frequency domain. zdivide(x, y) Return x/y, with 0 instead of NaN where y is 0. shrink1(x, alpha) Scalar shrinkage/soft thresholding function shrink2(x, alpha[, axis]) Vector shrinkage/soft thresholding function shrink12(x, alpha, beta[, axis]) Compound shrinkage/soft thresholding function [39] [10] proj_l2ball(b, s, r[, axes]) Project $$\mathbf{b}$$ into the $$\ell_2$$ ball of radius $$r$$ about $$\mathbf{s}$$, i.e. promote16(u[, fn]) Utility function for use with functions that do not support arrays of dtype np.float16. atleast_nd(n, u) If the input array has fewer than n dimensions, append singleton dimensions so that it is n dimensional. split(u[, axis]) Split an array into a list of arrays on the specified axis. blockcirculant(A) Construct a block circulant matrix from a tuple of arrays. fl2norm2(xf[, axis]) Compute the squared $$\ell_2$$ norm in the DFT domain, taking into account the unnormalised DFT scaling, i.e. rfl2norm2(xf, xs[, axis]) Compute the squared $$\ell_2$$ norm in the DFT domain, taking into account the unnormalised DFT scaling, i.e. rrs(ax, b) Compute relative residual $$\|\mathbf{b} - A \mathbf{x}\|_2 / \|\mathbf{b}\|_2$$ of the solution to a linear equation $$A \mathbf{x} = \mathbf{b}$$.

## Function Descriptions¶

sporco.linalg.complex_dtype(dtype)[source]

Construct the corresponding complex dtype for a given real dtype, e.g. the complex dtype corresponding to np.float32 is np.complex64.

Parameters: dtype : dtype A real dtype, e.g. np.float32, np.float64 cdtype : dtype The complex dtype corresponding to the input dtype
sporco.linalg.pyfftw_byte_aligned(array, dtype=None, n=None)[source]

Construct a byte-aligned array for efficient use by pyfftw. This function is a wrapper for pyfftw.byte_align

Parameters: array : ndarray Input array dtype : dtype, optional (default None) Output array dtype n : int, optional (default None) Output array should be aligned to n-byte boundary a : ndarray Array with required byte-alignment
sporco.linalg.pyfftw_empty_aligned(shape, dtype, order='C', n=None)

Construct an empty byte-aligned array for efficient use by pyfftw. This function is a wrapper for pyfftw.empty_aligned

Parameters: shape : sequence of ints Output array shape dtype : dtype Output array dtype order : {‘C’, ‘F’}, optional (default ‘C’) Specify whether arrays should be stored in row-major (C-style) or column-major (Fortran-style) order n : int, optional (default None) Output array should be aligned to n-byte boundary a : ndarray Empty array with required byte-alignment
sporco.linalg.pyfftw_rfftn_empty_aligned(shape, axes, dtype, order='C', n=None)

Construct an empty byte-aligned array for efficient use by pyfftw functions pyfftw.interfaces.numpy_fft.rfftn and pyfftw.interfaces.numpy_fft.irfftn. The shape of the empty array is appropriate for the output of pyfftw.interfaces.numpy_fft.rfftn applied to an array of the shape specified by parameter shape, and for the input of the corresponding pyfftw.interfaces.numpy_fft.irfftn call that reverses this operation.

Parameters: shape : sequence of ints Output array shape axes : sequence of ints Axes on which the FFT will be computed dtype : dtype Real dtype from which the complex dtype of the output array is derived order : {‘C’, ‘F’}, optional (default ‘C’) Specify whether arrays should be stored in row-major (C-style) or column-major (Fortran-style) order n : int, optional (default None) Output array should be aligned to n-byte boundary a : ndarray Empty array with required byte-alignment
sporco.linalg.fftn(a, s=None, axes=None, norm=None)[source]

Compute the multi-dimensional discrete Fourier transform. This function is a wrapper for pyfftw.interfaces.numpy_fft.fftn, with an interface similar to that of numpy.fft.fftn.

Parameters: a : array_like Input array (can be complex) s : sequence of ints, optional (default None) Shape of the output along each transformed axis (input is cropped or zero-padded to match). axes : sequence of ints, optional (default None) Axes over which to compute the DFT. af : complex ndarray DFT of input array
sporco.linalg.ifftn(a, s=None, axes=None, norm=None)[source]

Compute the multi-dimensional inverse discrete Fourier transform. This function is a wrapper for pyfftw.interfaces.numpy_fft.ifftn, with an interface similar to that of numpy.fft.ifftn.

Parameters: a : array_like Input array (can be complex) s : sequence of ints, optional (default None) Shape of the output along each transformed axis (input is cropped or zero-padded to match). axes : sequence of ints, optional (default None) Axes over which to compute the inverse DFT. af : complex ndarray Inverse DFT of input array
sporco.linalg.rfftn(a, s=None, axes=None, norm=None)[source]

Compute the multi-dimensional discrete Fourier transform for real input. This function is a wrapper for pyfftw.interfaces.numpy_fft.rfftn, with an interface similar to that of numpy.fft.rfftn.

Parameters: a : array_like Input array (taken to be real) s : sequence of ints, optional (default None) Shape of the output along each transformed axis (input is cropped or zero-padded to match). axes : sequence of ints, optional (default None) Axes over which to compute the DFT. af : complex ndarray DFT of input array
sporco.linalg.irfftn(a, s=None, axes=None, norm=None)[source]

Compute the inverse of the multi-dimensional discrete Fourier transform for real input. This function is a wrapper for pyfftw.interfaces.numpy_fft.irfftn, with an interface similar to that of numpy.fft.irfftn.

Parameters: a : array_like Input array s : sequence of ints Shape of the output along each transformed axis (input is cropped or zero-padded to match). This parameter is not optional because, unlike ifftn, the output shape cannot be uniquely determined from the input shape. axes : sequence of ints, optional (default None) Axes over which to compute the inverse DFT. af : ndarray Inverse DFT of input array
sporco.linalg.dctii(x, axes=None)[source]

Compute a multi-dimensional DCT-II over specified array axes. This function is implemented by calling the one-dimensional DCT-II scipy.fftpack.dct with normalization mode ‘ortho’ for each of the specified axes.

Parameters: a : array_like Input array axes : sequence of ints, optional (default None) Axes over which to compute the DCT-II. y : ndarray DCT-II of input array
sporco.linalg.idctii(x, axes=None)[source]

Compute a multi-dimensional inverse DCT-II over specified array axes. This function is implemented by calling the one-dimensional inverse DCT-II scipy.fftpack.idct with normalization mode ‘ortho’ for each of the specified axes.

Parameters: a : array_like Input array axes : sequence of ints, optional (default None) Axes over which to compute the inverse DCT-II. y : ndarray Inverse DCT-II of input array
sporco.linalg.fftconv(a, b, axes=(0, 1))[source]

Compute a multi-dimensional convolution via the Discrete Fourier Transform.

Parameters: a : array_like Input array b : array_like Input array axes : sequence of ints, optional (default (0, 1)) Axes on which to perform convolution ab : ndarray Convolution of input arrays, a and b, along specified axes
sporco.linalg.inner(x, y, axis=-1)[source]

Compute inner product of x and y on specified axis, equivalent to np.sum(x * y, axis=axis, keepdims=True).

Parameters: x : array_like Input array x y : array_like Input array y axis : int, optional (default -1) Axis over which to compute the sum y : ndarray Inner product array equivalent to summing x*y over the specified axis
sporco.linalg.dot(a, b, axis=-2)[source]

Compute the matrix product of a and the specified axes of b, with broadcasting over the remaining axes of b. This function is a generalisation of numpy.dot, supporting sum product over an arbitrary axis instead of just over the last axis.

If a and b are both 2D arrays, dot gives the same result as numpy.dot. If b has more than 2 axes, the result is obtained as follows (where a has shape (M0, M1) and b has shape (N0, N1, ..., M1, Nn, ...)):

1. Reshape a to shape ( 1,  1, ..., M0, M1,  1, ...)
2. Reshape b to shape (N0, N1, ...,  1, M1, Nn, ...)
3. Take the broadcast product and sum over the specified axis (the axis with dimension M1 in this example) to give an array of shape (N0, N1, ...,  M0,  1, Nn, ...)
4. Remove the singleton axis created by the summation to give an array of shape (N0, N1, ...,  M0, Nn, ...)
Parameters: a : array_like, 2D First component of product b : array_like, 2D or greater Second component of product axis : integer, optional (default -2) Axis of b over which sum is to be taken prod : ndarray Matrix product of a and specified axes of b, with broadcasting over the remaining axes of b
sporco.linalg.solvedbi_sm(ah, rho, b, c=None, axis=4)[source]

Solve a diagonal block linear system with a scaled identity term using the Sherman-Morrison equation.

The solution is obtained by independently solving a set of linear systems of the form (see [33])

$(\rho I + \mathbf{a} \mathbf{a}^H ) \; \mathbf{x} = \mathbf{b} \;\;.$

In this equation inner products and matrix products are taken along the specified axis of the corresponding multi-dimensional arrays; the solutions are independent over the other axes.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ rho : float Linear system parameter $$\rho$$ b : array_like Linear system component $$\mathbf{b}$$ c : array_like, optional (default None) Solution component $$\mathbf{c}$$ that may be pre-computed using solvedbi_sm_c and cached for re-use. axis : int, optional (default 4) Axis along which to solve the linear system x : ndarray Linear system solution $$\mathbf{x}$$
sporco.linalg.solvedbi_sm_c(ah, a, rho, axis=4)[source]

Compute cached component used by solvedbi_sm.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ a : array_like Linear system component $$\mathbf{a}$$ rho : float Linear system parameter $$\rho$$ axis : int, optional (default 4) Axis along which to solve the linear system c : ndarray Argument $$\mathbf{c}$$ used by solvedbi_sm
sporco.linalg.solvedbd_sm(ah, d, b, c=None, axis=4)[source]

Solve a diagonal block linear system with a diagonal term using the Sherman-Morrison equation.

The solution is obtained by independently solving a set of linear systems of the form (see [33])

$(\mathbf{d} + \mathbf{a} \mathbf{a}^H ) \; \mathbf{x} = \mathbf{b} \;\;.$

In this equation inner products and matrix products are taken along the specified axis of the corresponding multi-dimensional arrays; the solutions are independent over the other axes.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ d : array_like Linear system parameter $$\mathbf{d}$$ b : array_like Linear system component $$\mathbf{b}$$ c : array_like, optional (default None) Solution component $$\mathbf{c}$$ that may be pre-computed using solvedbd_sm_c and cached for re-use. axis : int, optional (default 4) Axis along which to solve the linear system x : ndarray Linear system solution $$\mathbf{x}$$
sporco.linalg.solvedbd_sm_c(ah, a, d, axis=4)[source]

Compute cached component used by solvedbd_sm.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ a : array_like Linear system component $$\mathbf{a}$$ d : array_like Linear system parameter $$\mathbf{d}$$ axis : int, optional (default 4) Axis along which to solve the linear system c : ndarray Argument $$\mathbf{c}$$ used by solvedbd_sm
sporco.linalg.solvemdbi_ism(ah, rho, b, axisM, axisK)[source]

Solve a multiple diagonal block linear system with a scaled identity term by iterated application of the Sherman-Morrison equation. The computation is performed in a way that avoids explictly constructing the inverse operator, leading to an $$O(K^2)$$ time cost.

The solution is obtained by independently solving a set of linear systems of the form (see [33])

$(\rho I + \mathbf{a}_0 \mathbf{a}_0^H + \mathbf{a}_1 \mathbf{a}_1^H + \; \ldots \; + \mathbf{a}_{K-1} \mathbf{a}_{K-1}^H) \; \mathbf{x} = \mathbf{b}$

where each $$\mathbf{a}_k$$ is an $$M$$-vector. The sums, inner products, and matrix products in this equation are taken along the M and K axes of the corresponding multi-dimensional arrays; the solutions are independent over the other axes.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ rho : float Linear system parameter $$\rho$$ b : array_like Linear system component $$\mathbf{b}$$ axisM : int Axis in input corresponding to index m in linear system axisK : int Axis in input corresponding to index k in linear system x : ndarray Linear system solution $$\mathbf{x}$$
sporco.linalg.solvemdbi_rsm(ah, rho, b, axisK, dimN=2)[source]

Solve a multiple diagonal block linear system with a scaled identity term by repeated application of the Sherman-Morrison equation. The computation is performed by explictly constructing the inverse operator, leading to an $$O(K)$$ time cost and $$O(M^2)$$ memory cost, where $$M$$ is the dimension of the axis over which inner products are taken.

The solution is obtained by independently solving a set of linear systems of the form (see [33])

$(\rho I + \mathbf{a}_0 \mathbf{a}_0^H + \mathbf{a}_1 \mathbf{a}_1^H + \; \ldots \; + \mathbf{a}_{K-1} \mathbf{a}_{K-1}^H) \; \mathbf{x} = \mathbf{b}$

where each $$\mathbf{a}_k$$ is an $$M$$-vector. The sums, inner products, and matrix products in this equation are taken along the M and K axes of the corresponding multi-dimensional arrays; the solutions are independent over the other axes.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ rho : float Linear system parameter $$\rho$$ b : array_like Linear system component $$\mathbf{b}$$ axisK : int Axis in input corresponding to index k in linear system dimN : int, optional (default 2) Number of spatial dimensions arranged as leading axes in input array. Axis M is taken to be at dimN+2. x : ndarray Linear system solution $$\mathbf{x}$$
sporco.linalg.solvemdbi_cg(ah, rho, b, axisM, axisK, tol=1e-05, mit=1000, isn=None)[source]

Solve a multiple diagonal block linear system with a scaled identity term using Conjugate Gradient (CG) via scipy.sparse.linalg.cg.

The solution is obtained by independently solving a set of linear systems of the form (see [33])

$(\rho I + \mathbf{a}_0 \mathbf{a}_0^H + \mathbf{a}_1 \mathbf{a}_1^H + \; \ldots \; + \mathbf{a}_{K-1} \mathbf{a}_{K-1}^H) \; \mathbf{x} = \mathbf{b}$

where each $$\mathbf{a}_k$$ is an $$M$$-vector. The inner products and matrix products in this equation are taken along the M and K axes of the corresponding multi-dimensional arrays; the solutions are independent over the other axes.

Parameters: ah : array_like Linear system component $$\mathbf{a}^H$$ rho : float Parameter rho b : array_like Linear system component $$\mathbf{b}$$ axisM : int Axis in input corresponding to index m in linear system axisK : int Axis in input corresponding to index k in linear system tol : float CG tolerance mit : int CG maximum iterations isn : array_like CG initial solution x : ndarray Linear system solution $$\mathbf{x}$$ cgit : int Number of CG iterations
sporco.linalg.lu_factor(A, rho, check_finite=True)[source]

Compute LU factorisation of either $$A^T A + \rho I$$ or $$A A^T + \rho I$$, depending on which matrix is smaller.

Parameters: A : array_like Array $$A$$ rho : float Scalar $$\rho$$ check_finite : bool, optional (default False) Flag indicating whether the input array should be checked for Inf and NaN values lu : ndarray Matrix containing U in its upper triangle, and L in its lower triangle, as returned by scipy.linalg.lu_factor piv : ndarray Pivot indices representing the permutation matrix P, as returned by scipy.linalg.lu_factor
sporco.linalg.lu_solve_ATAI(A, rho, b, lu, piv, check_finite=True)[source]

Solve the linear system $$(A^T A + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A^T A + \rho I)X = B$$ using scipy.linalg.lu_solve.

Parameters: A : array_like Matrix $$A$$ rho : float Scalar $$\rho$$ b : array_like Vector $$\mathbf{b}$$ or matrix $$B$$ lu : array_like Matrix containing U in its upper triangle, and L in its lower triangle, as returned by scipy.linalg.lu_factor piv : array_like Pivot indices representing the permutation matrix P, as returned by scipy.linalg.lu_factor check_finite : bool, optional (default False) Flag indicating whether the input array should be checked for Inf and NaN values x : ndarray Solution to the linear system
sporco.linalg.lu_solve_AATI(A, rho, b, lu, piv, check_finite=True)[source]

Solve the linear system $$(A A^T + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A A^T + \rho I)X = B$$ using scipy.linalg.lu_solve.

Parameters: A : array_like Matrix $$A$$ rho : float Scalar $$\rho$$ b : array_like Vector $$\mathbf{b}$$ or matrix $$B$$ lu : array_like Matrix containing U in its upper triangle, and L in its lower triangle, as returned by scipy.linalg.lu_factor piv : array_like Pivot indices representing the permutation matrix P, as returned by scipy.linalg.lu_factor check_finite : bool, optional (default False) Flag indicating whether the input array should be checked for Inf and NaN values x : ndarray Solution to the linear system
sporco.linalg.cho_factor(A, rho, lower=False, check_finite=True)[source]

Compute Cholesky factorisation of either $$A^T A + \rho I$$ or $$A A^T + \rho I$$, depending on which matrix is smaller.

Parameters: A : array_like Array $$A$$ rho : float Scalar $$\rho$$ lower : bool, optional (default False) Flag indicating whether lower or upper triangular factors are computed check_finite : bool, optional (default False) Flag indicating whether the input array should be checked for Inf and NaN values c : ndarray Matrix containing lower or upper triangular Cholesky factor, as returned by scipy.linalg.cho_factor lwr : bool Flag indicating whether the factor is lower or upper triangular
sporco.linalg.cho_solve_ATAI(A, rho, b, c, lwr, check_finite=True)[source]

Solve the linear system $$(A^T A + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A^T A + \rho I)X = B$$ using scipy.linalg.cho_solve.

Parameters: A : array_like Matrix $$A$$ rho : float Scalar $$\rho$$ b : array_like Vector $$\mathbf{b}$$ or matrix $$B$$ c : array_like Matrix containing lower or upper triangular Cholesky factor, as returned by scipy.linalg.cho_factor lwr : bool Flag indicating whether the factor is lower or upper triangular x : ndarray Solution to the linear system
sporco.linalg.cho_solve_AATI(A, rho, b, c, lwr, check_finite=True)[source]

Solve the linear system $$(A A^T + \rho I)\mathbf{x} = \mathbf{b}$$ or $$(A A^T + \rho I)X = B$$ using scipy.linalg.cho_solve.

Parameters: A : array_like Matrix $$A$$ rho : float Scalar $$\rho$$ b : array_like Vector $$\mathbf{b}$$ or matrix $$B$$ c : array_like Matrix containing lower or upper triangular Cholesky factor, as returned by scipy.linalg.cho_factor lwr : bool Flag indicating whether the factor is lower or upper triangular x : ndarray Solution to the linear system
sporco.linalg.zpad(x, pd, ax)[source]

Zero-pad array x with pd = (leading, trailing) zeros on axis ax.

Parameters: x : array_like Array to be padded pd : tuple Sequence of two ints (leading,trailing) specifying number of zeros for padding ax : int Axis to be padded xp : array_like Padded array
sporco.linalg.Gax(x, ax)[source]

Compute gradient of x along axis ax.

Parameters: x : array_like Input array ax : int Axis on which gradient is to be computed xg : ndarray Output array
sporco.linalg.GTax(x, ax)[source]

Compute transpose of gradient of x along axis ax.

Parameters: x : array_like Input array ax : int Axis on which gradient transpose is to be computed xg : ndarray Output array
sporco.linalg.GradientFilters(ndim, axes, axshp, dtype=None)[source]

Construct a set of filters for computing gradients in the frequency domain.

Parameters: ndim : integer Total number of dimensions in array in which gradients are to be computed axes : tuple of integers Axes on which gradients are to be computed axshp : tuple of integers Shape of axes on which gradients are to be computed dtype : dtype Data type of output arrays Gf : ndarray Frequency domain gradient operators $$\hat{G}_i$$ GHGf : ndarray Sum of products $$\sum_i \hat{G}_i^H \hat{G}_i$$
sporco.linalg.zdivide(x, y)[source]

Return x/y, with 0 instead of NaN where y is 0.

Parameters: x : array_like Numerator y : array_like Denominator z : ndarray Quotient x/y
sporco.linalg.shrink1(x, alpha)[source]

Scalar shrinkage/soft thresholding function

$\mathcal{S}_{1,\alpha}(\mathbf{x}) = \mathrm{sign}(\mathbf{x}) \odot \max(0, |\mathbf{x}| - \alpha) = \mathrm{prox}_f(\mathbf{x}) \;\; \text{where} \;\; f(\mathbf{u}) = \alpha \|\mathbf{u}\|_1 \;\;.$
Parameters: x : array_like Input array $$\mathbf{x}$$ alpha : float or array_like Shrinkage parameter $$\alpha$$ x : ndarray Output array
sporco.linalg.shrink2(x, alpha, axis=-1)[source]

Vector shrinkage/soft thresholding function

$\mathcal{S}_{2,\alpha}(\mathbf{x}) = \frac{\mathbf{x}}{\|\mathbf{x}\|_2} \max(0, \|\mathbf{x}\|_2 - \alpha) = \mathrm{prox}_f(\mathbf{x}) \;\; \text{where} \;\; f(\mathbf{u}) = \alpha \|\mathbf{u}\|_2 \;\;.$

The $$\ell_2$$ norm is applied over the specified axis of a multi-dimensional input (the last axis by default).

Parameters: x : array_like Input array $$\mathbf{x}$$ alpha : float or array_like Shrinkage parameter $$\alpha$$ axis : int, optional (default -1) Axis of x over which the $$\ell_2$$ norm x : ndarray Output array
sporco.linalg.shrink12(x, alpha, beta, axis=-1)[source]

Compound shrinkage/soft thresholding function [39] [10]

$\mathcal{S}_{1,2,\alpha,\beta}(\mathbf{x}) = \mathcal{S}_{2,\beta}(\mathcal{S}_{1,\alpha}(\mathbf{x})) = \mathrm{prox}_f(\mathbf{x}) \;\; \text{where} \;\; f(\mathbf{u}) = \alpha \|\mathbf{u}\|_1 + \beta \|\mathbf{u}\|_2 \;\;.$

The $$\ell_2$$ norm is applied over the specified axis of a multi-dimensional input (the last axis by default).

Parameters: x : array_like Input array $$\mathbf{x}$$ alpha : float or array_like Shrinkage parameter $$\alpha$$ beta : float or array_like Shrinkage parameter $$\beta$$ axis : int, optional (default -1) Axis of x over which the $$\ell_2$$ norm x : ndarray Output array
sporco.linalg.proj_l2ball(b, s, r, axes=None)[source]

Project $$\mathbf{b}$$ into the $$\ell_2$$ ball of radius $$r$$ about $$\mathbf{s}$$, i.e. $$\{ \mathbf{x} : \|\mathbf{x} - \mathbf{s} \|_2 \leq r \}$$.

Parameters: b : array_like Vector $$\mathbf{b}$$ to be projected s : array_like Centre of $$\ell_2$$ ball $$\mathbf{s}$$ r : float Radius of ball axes : sequence of ints, optional (default all axes) Axes over which to compute $$\ell_2$$ norms x : ndarray Projection of $$\mathbf{b}$$ into ball
sporco.linalg.promote16(u, fn=None, *args, **kwargs)[source]

Utility function for use with functions that do not support arrays of dtype np.float16. This function has two distinct modes of operation. If called with only the u parameter specified, the returned value is either u itself if u is not of dtype np.float16, or u promoted to np.float32 dtype if it is. If the function parameter fn is specified then u is conditionally promoted as described above, passed as the first argument to function fn, and the returned values are converted back to dtype np.float16 if u is of that dtype. Note that if parameter fn is specified, it may not be be specified as a keyword argument if it is followed by any non-keyword arguments.

Parameters: u : array_like Array to be promoted to np.float32 if it is of dtype np.float16 fn : function or None, optional (default None) Function to be called with promoted u as first parameter and *args and **kwargs as additional parameters *args Variable length list of arguments for function fn **kwargs Keyword arguments for function fn up : ndarray Conditionally dtype-promoted version of u if fn is None, or value(s) returned by fn, converted to the same dtype as u, if fn is a function
sporco.linalg.atleast_nd(n, u)[source]

If the input array has fewer than n dimensions, append singleton dimensions so that it is n dimensional. Note that the interface differs substantially from that of numpy.atleast_3d etc.

Parameters: n : int Minimum number of required dimensions u : array_like Input array v : ndarray Output array with at least n dimensions
sporco.linalg.split(u, axis=0)[source]

Split an array into a list of arrays on the specified axis. The length of the list is the shape of the array on the specified axis, and the corresponding axis is removed from each entry in the list. This function does not have the same behaviour as numpy.split.

Parameters: u : array_like Input array axis : int, optional (default 0) Axis on which to split the input array v : list of ndarray List of arrays
sporco.linalg.blockcirculant(A)[source]

Construct a block circulant matrix from a tuple of arrays. This is a block-matrix variant of scipy.linalg.circulant.

Parameters: A : tuple of array_like Tuple of arrays corresponding to the first block column of the output block matrix B : ndarray Output array
sporco.linalg.fl2norm2(xf, axis=(0, 1))[source]

Compute the squared $$\ell_2$$ norm in the DFT domain, taking into account the unnormalised DFT scaling, i.e. given the DFT of a multi-dimensional array computed via fftn, return the squared $$\ell_2$$ norm of the original array.

Parameters: xf : array_like Input array axis : sequence of ints, optional (default (0,1)) Axes on which the input is in the frequency domain x : float $$\|\mathbf{x}\|_2^2$$ where the input array is the result of applying fftn to the specified axes of multi-dimensional array $$\mathbf{x}$$
sporco.linalg.rfl2norm2(xf, xs, axis=(0, 1))[source]

Compute the squared $$\ell_2$$ norm in the DFT domain, taking into account the unnormalised DFT scaling, i.e. given the DFT of a multi-dimensional array computed via rfftn, return the squared $$\ell_2$$ norm of the original array.

Parameters: xf : array_like Input array xs : sequence of ints Shape of original array to which rfftn was applied to obtain the input array axis : sequence of ints, optional (default (0,1)) Axes on which the input is in the frequency domain x : float $$\|\mathbf{x}\|_2^2$$ where the input array is the result of applying rfftn to the specified axes of multi-dimensional array $$\mathbf{x}$$
sporco.linalg.rrs(ax, b)[source]

Compute relative residual $$\|\mathbf{b} - A \mathbf{x}\|_2 / \|\mathbf{b}\|_2$$ of the solution to a linear equation $$A \mathbf{x} = \mathbf{b}$$. Returns 1.0 if $$\mathbf{b} = 0$$.

Parameters: ax : array_like Linear component $$A \mathbf{x}$$ of equation b : array_like Constant component $$\mathbf{b}$$ of equation x : float Relative residual