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hw2.dvi

ECS130 Homework Assignment #2 Due: 4:00pm, Feburary 3, 2017

1. A real symmetric matrix A = AT is positive definite if any of the following equivalent condi-
tions hold:

• The quadratic form xTAx > 0 for all nonzero vectors x.

• All determinants formed from submatrices of any order centered on the diagonal of A
are positive.

• All eigenvalues of A are positive.

• There is a lower triangular matrix L such that A = LLT , called Cholesky decomposition
of A.

As you can see, the best way to check the positive definiteness is with Cholesky decomposition.

(a) Let n = 3 and write the formulas for computing the entries ℓij of L for a given 3 × 3
symmetric positive definite matrix A.

(b) Use the observations in (a) to derive formulas to compute the Cholesky decomposition
for an n× n symmetric positive definite (spd) matrix A.

(c) Program your formulas to compute the Cholesky decomposition of an n × n spd matrix
A. Check the correctness of your program by comparing with MATLAB’s built-in function
chol for the matrices A = (aij) with aij =

1

i+j−1
with n = 3, 4, 5.

2. Read section 2.9, and present your error analysis for the two “computed” solutions x̂1 =[
1.01
1.01

]
and x̂2 =

[
20.97

−18.99

]
of the linear system of equations

[
1000 999
999 998

]
x =

[
1999
1997

]
.

3. Assume you have a base-2 computer that stores floating-point numbers using a 6 bit normal-
ized mantissa and a 4 bit exponent, and a sign bit for each.

(a) For this machine, what is machine precision?

(b) What is the smallest positive normalized number that can be represented in this machine?

4. Consider the following program

x = 1;

delta = 1 / 2^(53);

for j = 1 : 2^(20),

x = x + delta;

end
By mathematical reasoning, what is the expected final value of x? Use your knowledge of
floating-point arithemtic to predict what it actually is. Verify by running the program and
explain the result.

5. Using mathematical reasoning, we know that for any positive number x, 2x is a number
greater than x. Is this true of floating-point numbers? Run the following program and

x = 1;

twox = 2*x;

k = 0;

while (twox > x)

1

x = twox;

twox = 2*x;

k = k + 1;

end

6. The polynomial p1(x) = (x − 1)
6 has the value zero at x = 1 and is positive elsewhere. The

expanded form of the polynomial

p2(x) = x
6
− 6×5 + 15×4 − 20×3 + 15×2 − 6x+ 1,

is mathematically equivalent. Plot p1(x) and p2(x) for 101 equally spaced points in the interval
[0.995, 1.005]. Explain the plots. (you should evaluate the polynomial p2(x) by Horner’s rule).

7. (a) Write a MATLAB function that computes the two roots of a quadratic polynomial
q(x) = x2 + bx+ c with good precision.

(b) Compare your computed results with MATLAB’s built-in function roots([a b c]) for
the following set of data:

1) b = −56, c = 1

2) b = −108, c = 1.

2

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