- Background In This Programming Assignment You Will Be Responsible For Implementing A Solver For The System Of Linear E 1 (48.99 KiB) Viewed 59 times
Vector. Can't use .dot but can use .transpose
Background: In this programming assignment, you will be responsible for implementing a solver for the system of linear equations Ax = where A is an n x n matrix whose columns are linearly independent XER" .BER" To implement the solver, you must apply the following theorem: THM | QR-Factorization If A e Fmxn matrix with linearly independent columns a, a, ... an. then there exists, 1. an m X n matrix Q whose columns ūū2, ..., ū are orthonormal, and 2. an n x n matrix R that is upper triangular and whose entries are defined by, rij = {fwa) for is; 0 for i>j such that A = QR. This referred to as the QR factorization (or decomposition) of matrix A. To find matrices Q and R from the QR Factorization Theorem, we apply Gram-Schimdt process to the columns of A. Then, • the columns of Q will be the orthonormal vectors u,u2, ..., un returned by the Gram Schimdt process, and • the entries rij of R will be computed using each column u as defined in the theorem. Luckily, you do not need to implement this process. A Python library called numpy contains a module called linalg with a function called or that returns the matrices Q and R in the QR factorization of a matrix A. Try running the following cell to see how it works.
Your Task: Assuming A E Rnxn is a Matrix object, and B ER" is a vec object, implement a function solve_gr(a, b) that uses the QR-factorization of A to compute and return the solution to the system Ax = 5.