Consider the 100 times 100 matrix A = (-4 1 1 1 - 4 1 1 1 - 4 1 1), b = (1 1), c

Question

Consider the 100 times 100 matrix A = (-4 1 1 1 - 4 1 1 1 - 4 1 1), b = (1 1), c =(0 1 2) which has -4 on the main diagonal and ones ons the 2nd and n/2-th sub/super diagonals (use integer division for n/2) (a) Solve the systems Ax = b and Ay = c. Calculate the LU-decomposition only once. Useful Python functions: scipy.linalg.lu factor, scipy.linalg.lu.solve (b) Calculate the LU-decomposition and plot the "sparsity patterns" of A, L and U. These are plots which are black in the positions where the respective matrices are non-zero and white where they are zero. What do you observe? Useful Python functions: scipy.linalg.lu, matplotlib.pyplot.spy (c) What is the difference between lu and lu_factor (In terms of the returned results)? Which one would you use in practice?

Explanation / Answer

In order to find all solutions to Ax = b we first check that the equation is

solvable, then find a particular solution. We get the complete solution of the

equation by adding the particular solution to all the vectors in the nullspace.

A particular solution

One way to find a particular solution to the equation Ax = b is to set all free

variables to zero, then solve for the pivot variables.

For our example matrix A, we let x2 = x4 = 0 to get the system of equations:

x1 + 2x3 = 1

2x3 = 3

1

which has the solution x3 = 3/2, x1 = 2. Our particular solution is:

2

0

xp = . 3/2

0

Combined with the nullspace

The general solution to Ax = b is given by xcomplete = xp + xn, where xn is a

generic vector in the nullspace. To see this, we add Axp

get A xp + xn = b for every vector xn in the nullspace.

= b to Axn = 0 and

Last lecture we learned that the nullspace of A is the collection of all combi­

2

1 2

0

nations of the special solutions and

. So the complete solution 0 2

0 1

1

to the equation Ax = 5 is:

6

2

0

3/2

+

2

1

0

2

+ c2

0

2

1

xcomplete = ,

0 0

where c1 and c2 are real numbers.

The nullspace of A is a two dimensional subspace of R4, and the solutions

2

0

to the equation Ax = b form a plane parallel to that through xp =

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