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Math 3607: Homework 4

2022

TOTAL: 30 points

• Problems marked with are to be done by hand; those marked with are to be solved using a computer.

• Important note. Do not use Symbolic Math Toolbox. Any work done using sym or syms will receive NO credit.

• Another important note. When asked write a MATLAB function, write one at the end of your live script.

1. (Proper usage of lu; FNC 2.6.1) Suppose that A P Rnˆn and b P Rn . On the left is correct MATLAB code to solve Ax “ b; on the right is similar but incorrect code. Explain using mathematical notation exactly what vector is found by the right-hand version.

[L ,U ] = lu (A );

x = U \ L \ b;

2. (FLOP counting; FNC 2.5.5) This problem is about evaluation of a polynomial

ppxq “ c1 ` c2x ` c3x2 ` ¨ ¨ ¨ ` cnxn´1 .

(a) Here is a little code to do the evaluation.

y = c (1);

xpow = 1;

for i = 2:n

xpow = xpow * x;

y = y + c (i)*xpow;

end

Assuming that x is a scalar, how many flops does this code take, as a function of n? Provide both the exact answer and the asymptotic answer as n Ñ 8.

Here is another code to do the same task.

y = c (n); % This algorithm is called Horner’s rule .

for j = n- 1:- 1:1

y = y *x + c (j);

end

Assuming that x is a scalar, how many flops does this code take, as a function of n? Provide both the exact answer and the asymptotic answer as n Ñ 8. Then compare the count to the one from (a).

3. (Understanding matrix multiplication) Do LM 12.5–3.

4. (Periodic fit; FNC 3.1.3) In this problem you are trying to find an approximation to the periodic function f ptq “ esinpt´1q over one period, 0 ď t ď 2π . In MATLAB, let t=linspace(0,2*pi,200)’ and let b be a column vector of evaluations of f at those points.