# pyerrors.linalg

View Source

    import numpy as np
    import autograd.numpy as anp  # Thinly-wrapped numpy
    from .obs import derived_observable, CObs, Obs, import_jackknife

    def matmul(*operands):
        """Matrix multiply all operands.

        Parameters
        ----------
        operands : numpy.ndarray
            Arbitrary number of 2d-numpy arrays which can be real or complex
            Obs valued.

        This implementation is faster compared to standard multiplication via the @ operator.
        """
        if any(isinstance(o[0, 0], CObs) for o in operands):
            extended_operands = []
            for op in operands:
                tmp = np.vectorize(lambda x: (np.real(x), np.imag(x)))(op)
                extended_operands.append(tmp[0])
                extended_operands.append(tmp[1])

            def multi_dot(operands, part):
                stack_r = operands[0]
                stack_i = operands[1]
                for op_r, op_i in zip(operands[2::2], operands[3::2]):
                    tmp_r = stack_r @ op_r - stack_i @ op_i
                    tmp_i = stack_r @ op_i + stack_i @ op_r

                    stack_r = tmp_r
                    stack_i = tmp_i

                if part == 'Real':
                    return stack_r
                else:
                    return stack_i

            def multi_dot_r(operands):
                return multi_dot(operands, 'Real')

            def multi_dot_i(operands):
                return multi_dot(operands, 'Imag')

            Nr = derived_observable(multi_dot_r, extended_operands, array_mode=True)
            Ni = derived_observable(multi_dot_i, extended_operands, array_mode=True)

            res = np.empty_like(Nr)
            for (n, m), entry in np.ndenumerate(Nr):
                res[n, m] = CObs(Nr[n, m], Ni[n, m])

            return res
        else:
            def multi_dot(operands):
                stack = operands[0]
                for op in operands[1:]:
                    stack = stack @ op
                return stack
            return derived_observable(multi_dot, operands, array_mode=True)

    def jack_matmul(*operands):
        """Matrix multiply both operands making use of the jackknife approximation.

        Parameters
        ----------
        operands : numpy.ndarray
            Arbitrary number of 2d-numpy arrays which can be real or complex
            Obs valued.

        For large matrices this is considerably faster compared to matmul.
        """

        def _exp_to_jack(matrix):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = entry.export_jackknife()
            return base_matrix

        def _imp_from_jack(matrix, name, idl):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = import_jackknife(entry, name, [idl])
            return base_matrix

        def _exp_to_jack_c(matrix):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = entry.real.export_jackknife() + 1j * entry.imag.export_jackknife()
            return base_matrix

        def _imp_from_jack_c(matrix, name, idl):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = CObs(import_jackknife(entry.real, name, [idl]),
                                          import_jackknife(entry.imag, name, [idl]))
            return base_matrix

        if any(isinstance(o.flat[0], CObs) for o in operands):
            name = operands[0].flat[0].real.names[0]
            idl = operands[0].flat[0].real.idl[name]

            r = _exp_to_jack_c(operands[0])
            for op in operands[1:]:
                if isinstance(op.flat[0], CObs):
                    r = r @ _exp_to_jack_c(op)
                else:
                    r = r @ op
            return _imp_from_jack_c(r, name, idl)
        else:
            name = operands[0].flat[0].names[0]
            idl = operands[0].flat[0].idl[name]

            r = _exp_to_jack(operands[0])
            for op in operands[1:]:
                if isinstance(op.flat[0], Obs):
                    r = r @ _exp_to_jack(op)
                else:
                    r = r @ op
            return _imp_from_jack(r, name, idl)

    def einsum(subscripts, *operands):
        """Wrapper for numpy.einsum

        Parameters
        ----------
        subscripts : str
            Subscripts for summation (see numpy documentation for details)
        operands : numpy.ndarray
            Arbitrary number of 2d-numpy arrays which can be real or complex
            Obs valued.
        """

        def _exp_to_jack(matrix):
            base_matrix = []
            for index, entry in np.ndenumerate(matrix):
                base_matrix.append(entry.export_jackknife())
            return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)

        def _exp_to_jack_c(matrix):
            base_matrix = []
            for index, entry in np.ndenumerate(matrix):
                base_matrix.append(entry.real.export_jackknife() + 1j * entry.imag.export_jackknife())
            return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)

        def _imp_from_jack(matrix, name, idl):
            base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
            for index in np.ndindex(matrix.shape[:-1]):
                base_matrix[index] = import_jackknife(matrix[index], name, [idl])
            return base_matrix

        def _imp_from_jack_c(matrix, name, idl):
            base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
            for index in np.ndindex(matrix.shape[:-1]):
                base_matrix[index] = CObs(import_jackknife(matrix[index].real, name, [idl]),
                                          import_jackknife(matrix[index].imag, name, [idl]))
            return base_matrix

        for op in operands:
            if isinstance(op.flat[0], CObs):
                name = op.flat[0].real.names[0]
                idl = op.flat[0].real.idl[name]
                break
            elif isinstance(op.flat[0], Obs):
                name = op.flat[0].names[0]
                idl = op.flat[0].idl[name]
                break

        conv_operands = []
        for op in operands:
            if isinstance(op.flat[0], CObs):
                conv_operands.append(_exp_to_jack_c(op))
            elif isinstance(op.flat[0], Obs):
                conv_operands.append(_exp_to_jack(op))
            else:
                conv_operands.append(op)

        tmp_subscripts = ','.join([o + '...' for o in subscripts.split(',')])
        extended_subscripts = '->'.join([o + '...' for o in tmp_subscripts.split('->')[:-1]] + [tmp_subscripts.split('->')[-1]])
        einsum_path = np.einsum_path(extended_subscripts, *conv_operands, optimize='optimal')[0]
        jack_einsum = np.einsum(extended_subscripts, *conv_operands, optimize=einsum_path)

        if jack_einsum.dtype == complex:
            result = _imp_from_jack_c(jack_einsum, name, idl)
        elif jack_einsum.dtype == float:
            result = _imp_from_jack(jack_einsum, name, idl)
        else:
            raise Exception("Result has unexpected datatype")

        if result.shape == ():
            return result.flat[0]
        else:
            return result

    def inv(x):
        """Inverse of Obs or CObs valued matrices."""
        return _mat_mat_op(anp.linalg.inv, x)

    def cholesky(x):
        """Cholesky decomposition of Obs or CObs valued matrices."""
        return _mat_mat_op(anp.linalg.cholesky, x)

    def det(x):
        """Determinant of Obs valued matrices."""
        return _scalar_mat_op(anp.linalg.det, x)

    def _scalar_mat_op(op, obs, **kwargs):
        """Computes the matrix to scalar operation op to a given matrix of Obs."""
        def _mat(x, **kwargs):
            dim = int(np.sqrt(len(x)))

            mat = []
            for i in range(dim):
                row = []
                for j in range(dim):
                    row.append(x[j + dim * i])
                mat.append(row)

            return op(anp.array(mat))

        if isinstance(obs, np.ndarray):
            raveled_obs = (1 * (obs.ravel())).tolist()
        else:
            raise TypeError('Unproper type of input.')
        return derived_observable(_mat, raveled_obs, **kwargs)

    def _mat_mat_op(op, obs, **kwargs):
        """Computes the matrix to matrix operation op to a given matrix of Obs."""
        # Use real representation to calculate matrix operations for complex matrices
        if isinstance(obs.ravel()[0], CObs):
            A = np.empty_like(obs)
            B = np.empty_like(obs)
            for (n, m), entry in np.ndenumerate(obs):
                if hasattr(entry, 'real') and hasattr(entry, 'imag'):
                    A[n, m] = entry.real
                    B[n, m] = entry.imag
                else:
                    A[n, m] = entry
                    B[n, m] = 0.0
            big_matrix = np.block([[A, -B], [B, A]])
            op_big_matrix = derived_observable(lambda x, **kwargs: op(x), [big_matrix], array_mode=True)[0]
            dim = op_big_matrix.shape[0]
            op_A = op_big_matrix[0: dim // 2, 0: dim // 2]
            op_B = op_big_matrix[dim // 2:, 0: dim // 2]
            res = np.empty_like(op_A)
            for (n, m), entry in np.ndenumerate(op_A):
                res[n, m] = CObs(op_A[n, m], op_B[n, m])
            return res
        else:
            return derived_observable(lambda x, **kwargs: op(x), [obs], array_mode=True)[0]

    def eigh(obs, **kwargs):
        """Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh."""
        w = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[0], obs)
        v = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[1], obs)
        return w, v

    def eig(obs, **kwargs):
        """Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig."""
        w = derived_observable(lambda x, **kwargs: anp.real(anp.linalg.eig(x)[0]), obs)
        return w

    def pinv(obs, **kwargs):
        """Computes the Moore-Penrose pseudoinverse of a matrix of Obs."""
        return derived_observable(lambda x, **kwargs: anp.linalg.pinv(x), obs)

    def svd(obs, **kwargs):
        """Computes the singular value decomposition of a matrix of Obs."""
        u = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[0], obs)
        s = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[1], obs)
        vh = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[2], obs)
        return (u, s, vh)

def matmul(*operands):
View Source

    def matmul(*operands):
        """Matrix multiply all operands.

        Parameters
        ----------
        operands : numpy.ndarray
            Arbitrary number of 2d-numpy arrays which can be real or complex
            Obs valued.

        This implementation is faster compared to standard multiplication via the @ operator.
        """
        if any(isinstance(o[0, 0], CObs) for o in operands):
            extended_operands = []
            for op in operands:
                tmp = np.vectorize(lambda x: (np.real(x), np.imag(x)))(op)
                extended_operands.append(tmp[0])
                extended_operands.append(tmp[1])

            def multi_dot(operands, part):
                stack_r = operands[0]
                stack_i = operands[1]
                for op_r, op_i in zip(operands[2::2], operands[3::2]):
                    tmp_r = stack_r @ op_r - stack_i @ op_i
                    tmp_i = stack_r @ op_i + stack_i @ op_r

                    stack_r = tmp_r
                    stack_i = tmp_i

                if part == 'Real':
                    return stack_r
                else:
                    return stack_i

            def multi_dot_r(operands):
                return multi_dot(operands, 'Real')

            def multi_dot_i(operands):
                return multi_dot(operands, 'Imag')

            Nr = derived_observable(multi_dot_r, extended_operands, array_mode=True)
            Ni = derived_observable(multi_dot_i, extended_operands, array_mode=True)

            res = np.empty_like(Nr)
            for (n, m), entry in np.ndenumerate(Nr):
                res[n, m] = CObs(Nr[n, m], Ni[n, m])

            return res
        else:
            def multi_dot(operands):
                stack = operands[0]
                for op in operands[1:]:
                    stack = stack @ op
                return stack
            return derived_observable(multi_dot, operands, array_mode=True)

Matrix multiply all operands.

###### Parameters

- operands (numpy.ndarray): Arbitrary number of 2d-numpy arrays which can be real or complex Obs valued.
- This implementation is faster compared to standard multiplication via the @ operator.

def jack_matmul(*operands):
View Source

    def jack_matmul(*operands):
        """Matrix multiply both operands making use of the jackknife approximation.

        Parameters
        ----------
        operands : numpy.ndarray
            Arbitrary number of 2d-numpy arrays which can be real or complex
            Obs valued.

        For large matrices this is considerably faster compared to matmul.
        """

        def _exp_to_jack(matrix):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = entry.export_jackknife()
            return base_matrix

        def _imp_from_jack(matrix, name, idl):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = import_jackknife(entry, name, [idl])
            return base_matrix

        def _exp_to_jack_c(matrix):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = entry.real.export_jackknife() + 1j * entry.imag.export_jackknife()
            return base_matrix

        def _imp_from_jack_c(matrix, name, idl):
            base_matrix = np.empty_like(matrix)
            for index, entry in np.ndenumerate(matrix):
                base_matrix[index] = CObs(import_jackknife(entry.real, name, [idl]),
                                          import_jackknife(entry.imag, name, [idl]))
            return base_matrix

        if any(isinstance(o.flat[0], CObs) for o in operands):
            name = operands[0].flat[0].real.names[0]
            idl = operands[0].flat[0].real.idl[name]

            r = _exp_to_jack_c(operands[0])
            for op in operands[1:]:
                if isinstance(op.flat[0], CObs):
                    r = r @ _exp_to_jack_c(op)
                else:
                    r = r @ op
            return _imp_from_jack_c(r, name, idl)
        else:
            name = operands[0].flat[0].names[0]
            idl = operands[0].flat[0].idl[name]

            r = _exp_to_jack(operands[0])
            for op in operands[1:]:
                if isinstance(op.flat[0], Obs):
                    r = r @ _exp_to_jack(op)
                else:
                    r = r @ op
            return _imp_from_jack(r, name, idl)

Matrix multiply both operands making use of the jackknife approximation.

###### Parameters

- operands (numpy.ndarray): Arbitrary number of 2d-numpy arrays which can be real or complex Obs valued.
- For large matrices this is considerably faster compared to matmul.

def einsum(subscripts, *operands):
View Source

    def einsum(subscripts, *operands):
        """Wrapper for numpy.einsum

        Parameters
        ----------
        subscripts : str
            Subscripts for summation (see numpy documentation for details)
        operands : numpy.ndarray
            Arbitrary number of 2d-numpy arrays which can be real or complex
            Obs valued.
        """

        def _exp_to_jack(matrix):
            base_matrix = []
            for index, entry in np.ndenumerate(matrix):
                base_matrix.append(entry.export_jackknife())
            return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)

        def _exp_to_jack_c(matrix):
            base_matrix = []
            for index, entry in np.ndenumerate(matrix):
                base_matrix.append(entry.real.export_jackknife() + 1j * entry.imag.export_jackknife())
            return np.asarray(base_matrix).reshape(matrix.shape + base_matrix[0].shape)

        def _imp_from_jack(matrix, name, idl):
            base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
            for index in np.ndindex(matrix.shape[:-1]):
                base_matrix[index] = import_jackknife(matrix[index], name, [idl])
            return base_matrix

        def _imp_from_jack_c(matrix, name, idl):
            base_matrix = np.empty(shape=matrix.shape[:-1], dtype=object)
            for index in np.ndindex(matrix.shape[:-1]):
                base_matrix[index] = CObs(import_jackknife(matrix[index].real, name, [idl]),
                                          import_jackknife(matrix[index].imag, name, [idl]))
            return base_matrix

        for op in operands:
            if isinstance(op.flat[0], CObs):
                name = op.flat[0].real.names[0]
                idl = op.flat[0].real.idl[name]
                break
            elif isinstance(op.flat[0], Obs):
                name = op.flat[0].names[0]
                idl = op.flat[0].idl[name]
                break

        conv_operands = []
        for op in operands:
            if isinstance(op.flat[0], CObs):
                conv_operands.append(_exp_to_jack_c(op))
            elif isinstance(op.flat[0], Obs):
                conv_operands.append(_exp_to_jack(op))
            else:
                conv_operands.append(op)

        tmp_subscripts = ','.join([o + '...' for o in subscripts.split(',')])
        extended_subscripts = '->'.join([o + '...' for o in tmp_subscripts.split('->')[:-1]] + [tmp_subscripts.split('->')[-1]])
        einsum_path = np.einsum_path(extended_subscripts, *conv_operands, optimize='optimal')[0]
        jack_einsum = np.einsum(extended_subscripts, *conv_operands, optimize=einsum_path)

        if jack_einsum.dtype == complex:
            result = _imp_from_jack_c(jack_einsum, name, idl)
        elif jack_einsum.dtype == float:
            result = _imp_from_jack(jack_einsum, name, idl)
        else:
            raise Exception("Result has unexpected datatype")

        if result.shape == ():
            return result.flat[0]
        else:
            return result

Wrapper for numpy.einsum

###### Parameters

- subscripts (str): Subscripts for summation (see numpy documentation for details)
- operands (numpy.ndarray): Arbitrary number of 2d-numpy arrays which can be real or complex Obs valued.

def inv(x):
View Source

    def inv(x):
        """Inverse of Obs or CObs valued matrices."""
        return _mat_mat_op(anp.linalg.inv, x)

Inverse of Obs or CObs valued matrices.

def cholesky(x):
View Source

    def cholesky(x):
        """Cholesky decomposition of Obs or CObs valued matrices."""
        return _mat_mat_op(anp.linalg.cholesky, x)

Cholesky decomposition of Obs or CObs valued matrices.

def det(x):
View Source

    def det(x):
        """Determinant of Obs valued matrices."""
        return _scalar_mat_op(anp.linalg.det, x)

Determinant of Obs valued matrices.

def eigh(obs, **kwargs):
View Source

    def eigh(obs, **kwargs):
        """Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh."""
        w = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[0], obs)
        v = derived_observable(lambda x, **kwargs: anp.linalg.eigh(x)[1], obs)
        return w, v

Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

def eig(obs, **kwargs):
View Source

    def eig(obs, **kwargs):
        """Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig."""
        w = derived_observable(lambda x, **kwargs: anp.real(anp.linalg.eig(x)[0]), obs)
        return w

Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.

def pinv(obs, **kwargs):
View Source

    def pinv(obs, **kwargs):
        """Computes the Moore-Penrose pseudoinverse of a matrix of Obs."""
        return derived_observable(lambda x, **kwargs: anp.linalg.pinv(x), obs)

Computes the Moore-Penrose pseudoinverse of a matrix of Obs.

def svd(obs, **kwargs):
View Source

    def svd(obs, **kwargs):
        """Computes the singular value decomposition of a matrix of Obs."""
        u = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[0], obs)
        s = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[1], obs)
        vh = derived_observable(lambda x, **kwargs: anp.linalg.svd(x, full_matrices=False)[2], obs)
        return (u, s, vh)

Computes the singular value decomposition of a matrix of Obs.