Source code for cla_utils.exercises3

import numpy as np


def householder(A):
    """
    Given a real mxn matrix A, find the reduction to upper triangular matrix R
    using Householder transformations. The reduction should be done "in-place",
    so that A is transformed to R.

    :param A: an mxn-dimensional numpy array
    """

    m, n = A.shape

    raise NotImplementedError


def solve_U(U, b):
    """
    Solve systems Ux_i=b_i for x_i with U upper triangular, i=1,2,...,k

    :param U: an mxm-dimensional numpy array, assumed upper triangular
    :param b: an mxk-dimensional numpy array, with ith column containing 
       b_i
    :return x: an mxk-dimensional numpy array, with ith column containing 
       the solution x_i

    """
                     
    raise NotImplementedError


def householder_solve(A, b):
    """
    Given a real mxm matrix A, use the Householder transformation to solve
    Ax_i=b_i, i=1,2,...,k.

    :param A: an mxm-dimensional numpy array
    :param b: an mxk-dimensional numpy array whose columns are the \
    right-hand side vectors b_1,b_2,...,b_k.

    :return x: an mxk-dimensional numpy array whose columns are the \
    right-hand side vectors x_1,x_2,...,x_k.
    """

    raise NotImplementedError

    return x


def householder_qr(A):
    """
    Given a real mxn matrix A, use the Householder transformation to find
    the full QR factorisation of A.

    :param A: an mxn-dimensional numpy array

    :return Q: an mxm-dimensional numpy array
    :return R: an mxn-dimensional numpy array
    """

    raise NotImplementedError

    return Q, R


def householder_ls(A, b):
    """
    Given a real mxn matrix A and an m dimensional vector b, find the
    least squares solution to Ax = b.

    :param A: an mxn-dimensional numpy array
    :param b: an m-dimensional numpy array

    :return x: an n-dimensional numpy array
    """

    raise NotImplementedError

    return x