SVD: A concrete example

Usage notes

Given the following matrix and vector:

The objective of this exercise is to find a vector x so that the residue r = Ax - b is of minimal norm while

a)

Fill in the following blanks:
Solve the ___________________ problem to figure out the cut-point of the ___________________ so that the norm of the solution is less or equal to 5.

The Maple command Svd(A,U,V) provide us the matrices U,

and V such that

Applying this command to our data and then computing

and rounding everything to three significant digits, we get:

b)

For each of the following cases, find a solution x and its norm. Do not use Maple, all values can be computed with the pocket calculator. To enter the vector x, use Maple syntax. When you enter the results, the values must be rounded to two significant digits.

i) using all singular values.

x = ___________________ ||x|| = ___________________

ii) discarding one singular value.

x = ___________________ ||x|| = ___________________

iii) discarding two singular values.

x = ___________________ ||x|| = ___________________

c)

To keep

, the number of singular values to use is ___________________ The corresponding solution is x = ___________________

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