代做Math 373-Fall、代写Matlab语言、代写sup-norm、代做Matlab设计

日期：2018-12-04 10:18

Math 373-Fall 2018 Midterm Project 4 Due: 6:15pm, 12 December

Work through each of the following problems. Be sure to follow the midterm project

guidelines found in the resources section on sakai. Turn in all coding portions digitally and

the all analysis portions in class. Both portions must be turned in by the due date and

time.

Throughout this project the norm of a vector in R

n should be taken to mean the sup-norm i.e. if x ∈ R

n,

then

||x|| := ||x||∞ = max {|x1| , . . . , |xn|} .

1. (10 points) Further adventures in root finding

For this project you will need to expand your findroot function from previous projects to higher

dimensional vector fields. For a function, f : R

n → R

n with n ≥ 1, it should be capable of the following

calls (in addition to the functionality from previous projects).

findroot(f, Df, x0) ← x attempts to return x ∈ R

n satisfying  = ||Af(x)|| < 10?10 by Newton

iteration with initial guess, x0 ∈ R

n using no more than N = 100 iterations where A ≈ Df ?1

(x)

is nonsingular. Otherwise, it should return x = NaN. Here f and Df are function handles for a

function and its derivative. If an approximate root is found, it should do 5 additional iterations

before returning.

findroot(f, x0) ← x attempts to return a root as in the 1st case without supplying the derivative.

Instead, this should be approximated by finite differences.

Use findroot to answer the following questions. You may also use the builtin functions polyval, and

conv if needed.

(a) Write a script called check findroot which uses findroot to find the maximum value of the

function f(x) = x1 + 2x2 + 3x3 defined on R

3

, subject to the constraints:

x1 x2 + x3 = 1 x

2

1 + x

2

2 = 1.

For this example, you should specify the derivative in your calls to findroot. Then, find this

maximum value by hand to verify that findroot is implemented correctly.

(b) Write a MATLAB function called taylor step which uses the Taylor method to integrate the

following scalar differential equation

x˙ = x

2 2x x(0) = x0 ∈ R.

for a single time step. It should be capable of the following call

taylor step(x0, N) ← (a, tf ) returns the first N terms of the Taylor transform of x as a vector of

the form

a = (a0, a1, . . . , aN?1).

and tf is an estimate on the radius of convergence of this power series.

The value of t should be estimated using one of the following two methods

Let tf = max 

t > 0 : |aN1|t

N1 < μ(1)

where μ(1) denotes the machine unit.

Or tf is the slope of the line obtained by fitting {log(a0), log(a1), . . . , log(aN1)} by linear least

squares regression.

The integrator should use the recursive formula obtained from the differential equation to compute

the first 5 Taylor coefficients. These should be used as an initial guess for Newton’s method which

is used to compute the rest of the coefficients.

(c) Write a MATLAB script called taylor ivp which calls taylor step for multiple time steps to solve

the initial value problem on the interval t ∈ [0, 3] for x0 ∈ {1.99, 2.001} using N ∈ {5, 10, 20, 30, 50, 200}.

Plot each of these solution in the same axis. What impact does increasing N have on the solution?

(d) Write a MATLAB script called taylor ivp v2 which calls taylor step for multiple time steps to

solve the initial value problem on the interval t ∈ [0, 3] for x0 = 2.01 using N = 200. Analyze the

output of this script and the differential equation to explain mathematically what is happening.

2. (8 points) Numerical methods for Fourier series

For this exercise you will need to write MATLAB functions called FFT and iFFT which are capable of

the following calls.

FFT(x) ← X returns the discrete Fourier transform of the vector x ∈ C

n.

iFFT(X) ← x returns the inverse discrete Fourier transform of the vector X ∈ C

n.

Both algorithms should have computational complexity on order of O(n log n). Use these functions to

complete the following exercises.

(a) Fix a constant 0 < L < ∞ and define the set

B =



1

2L

, cos(πx

L

),sin(πx

L

), cos(2πx

L

),sin(2πx

L

), . . . , cos( jπx

L

),sin( jπx

L

), . . . 

.

Show that B is an orthonormal set of vectors in C

(0)([L, L]) with respect to the inner product

hf, gi =

1

L

Z L

?L

f(x)g(x) dx.

(b) Use the result in part (a) to compute the Fourier series for the function f(x) = x defined on [?π, π]

by hand. Then write a script called check FFT which uses this computation to verify that your FFT

and iFFT implementations are correct.

(c) Consider the following differential equation (known as the Vanderpol oscillator)

x˙ =



x2

μ(1 x

2

1

)x2 x1



x ∈ R

2

where μ > 0 is a parameter. This equation is well studied and it is known that for any choice of

μ, it has an unstable equilibrium at the origin and a globally attracting periodic orbit. For μ = 1,

this vector field is builtin to MATLAB in a function called vdp1.

Write a script called vdp periodic orbit which uses the functions FFT, iFFT, and findroot to

compute a trigonometric polynomial which approximates the Fourier series for this periodic orbit

when μ = 1. You may use any of the numerical methods implemented in previous projects to help

estimate the orbit, its period, or any other investigations which you may find helpful.

3. (7 points) A square root solver with verified error bounds

Let c be a positive real number and √

c denotes the unique positive root of the function f(x) = x

2 c.

(a) Show that the Newton map for f is Lipschitz with constant k < 1

2

on the domain D = (pc

2

, ∞).

Hint: Use the mean value inequality.

(b) Write a MATLAB function called verified sqrt which is capable of the following call

verified sqrt(c) ← (x, d) where x ≈

√

c is obtained by calling findroot and d is a number of

digits guaranteed to be correct via the contraction mapping theorem i.e. x satisfies the error

bound ≤ 10d

Write a script called check sqrt which uses this function to compute √

2 and compare it to the

MATLAB builtin value up to d decimal places. Then answer the following questions.

Assuming the MATLAB builtin is accurate to working precision, explain why these values must

match to at least d decimal places.

Is it possible for them to match to more than d decimal places? Explain why or why not.

(c) Write a MATLAB script called compare error which uses calls to verified sqrt to approximate

√

x for each x ∈



10k

: k = 1, 5, 10, 15, . . . , 50

. Compare the number of digits of accuracy

output by your function with the actual values output for √

x.

For which values of x does your function output an approximation and a “guaranteed” number of

decimal digits which is actually false. Explain why this does not violate the contraction mapping

theorem.