How do you write a Matlab function that determines the case of the soluion to Ax = b.

Write a MATLAB function called LeastSquares that determines the case of the solution to Ax = b. In addition, the function also solves for x accordingly and returns the norm of the error vector. 
The function header should look like the following:
function [flag xstar cstar] = LeastSquares(A,b,method)
The inputs to this function are defined as follows:
 A: a matrix of arbitrary size.
 b: a column vector, which has the same number of rows as matrix A.
 method: should be filled with either 'mldivide' or 'pseudoinverse'. The method that you specify will be used to solve for the cases where non-unique solutions exist. If method is not specified, the default method should be 'mldivide'.
The outputs for this function are as follows:
 xstar: the solution to Ax = b.
 cstar: the norm of the error vector.
 flag is a string, with the following values, depending on the type of solution:
{ Unique Exact Solution: flag is '1-Unique Exact Solution'
{ Non-Unique Exact Solutions: flag is '2-Non-Unique Exact Solutions'
{ Unique Least Squares Solution: flag is '3-Unique Least Squares Solution'
{ Non-Unique Least Squares Solutions: flag is '4-Non-Unique Least Squares Solutions'

The test cases I have been given are as follows:

    % Test Case
    >> A = [1 2 3;2 4 8;3 5 7];
    >> b = [22;54;53];
    >> [flag xstar cstar] = LeastSquares(A,b)
    flag =
    1-Unique Exact Solution
    xstar =
    1.0000
    3.0000
    5.0000
    cstar =
    0
    >> A = [1 5;2 10;3 15];
    >> b = [10;20;30];
    >> [flag xstar cstar] = LeastSquares(A,b,'mldivide')
    flag =
    2-Non-Unique Exact Solutions
    xstar =
    0
    2.0000
    cstar =
    1.6281e-014

    >> A = [1 2 3;2 4 6;3 6 9];
    >> b = [22;54;77];
    >> [flag xstar cstar] = LeastSquares(A,b,'pseudoinverse')
    flag=
    4-Non-Unique Least Squares Solutions
    xstar =
    1.8418
    3.6837
    5.5255
    cstar =
    4.5119

Any help would be greatly appreciated. Thank you.

[Walter Roberson]

Please ask a specific question, as otherwise we will just refer you to the documentation on how to program in MATLAB.

Answer 1

[Matt J]

Hints:

• Use rank(A) to determine whether a solution is unique.
• Use rank([A,b]) to determine whether it is exact.