MAT 128B, Winter 2019
Programming Project 1
(due by Wednesday, January 30, 11:59 pm)
General Instructions
You are required to submit each of your programming projects via file upload to Canvas.
Note that the due dates set in Canvas are hard deadlines. I will not accept any submissions
outside of Canvas or after the deadline.
Write a report that includes all required numerical results, a discussion of your results, and
explanations of runs for which a method failed. Your report should be at least one page long,
but not longer than three pages.
When you are asked to print out numerical results, print numbers in 16-digit floating-point format.
You can use the Matlab command “format long e” to switch to that format from Matlab’s
default. For example, the number 10π would be printed out as 3.141592653589793e+01
in 16-digit floating-point format.
For each programming project, upload these files: a single pdf file of your report and a
complete set of Matlab files that lets us run and check your programs. This set should consist
of one file for each Matlab function you are asked to write and a single driver file for each
of the problems that you are asked to test your programs on. Each of these driver files
should generate the required numerical results for all runs of a problem. So for this current
project, you should submit a total of 7 Matlab files: 4 files for the functions FPI, NEWTON,
DAMPED NEWTON, and SECANT and three driver files for Problems 1–3.
We consider the problem of computing solutions of nonlinear equations
f(x) = 0, (1)
where f : R 7→ R is a continuous function. A solution x of (1) is called a root of f. Newton’s
method, Newton’s method with damping, and the secant method are root-finding procedures that
are applied directly to the function f. Fixed-point iteration is applied to a continuous function g
the fixed points of which are roots of f.
Write Matlab functions FPI, NEWTON, DAMPED NEWTON, and SECANT for carrying out the
versions of fixed-point iteration, Newton’s method, Newton’s method with damping, and the secant
method that were presented in class. For Newton’s method and Newton’s method with damping,
the function f is assumed to be continuously differentiable.
For all 4 functions, use the stopping criterion
|xi+1 ? xi
|
max{ |xi+1|, 1 }
< TOL.
Here, xi+1 and xi denotes the approximate root of f generated in the current iteration and previous
iteration, respectively. Make an effort to write your programs such that you use as few function
evaluations (of g, f, and f
0
) as possible.
The input parameters for FPI should be:
A Matlab function for evaluating the function g at any given x
The initial guess x0
The convergence tolerance TOL
The integer N0 to safeguard against infinite loops due to bugs or non-converging iterates
The input parameters for NEWTON should be:
A Matlab function for evaluating the function f at any given x
A Matlab function for evaluating the derivative f
0 at any given x
The initial guess x0
The convergence tolerance TOL
The integer N0 to safeguard against infinite loops due to bugs or non-converging iterates
The input parameters for DAMPED NEWTON should be:
A Matlab function for evaluating the function f at any given x
A Matlab function for evaluating the derivative f
0 at any given x
The initial guess x0
The convergence tolerance TOL
The integer N0 to safeguard against infinite loops due to bugs or non-converging iterates
A small λmin > 0 to safeguard against tiny λi
’s
The input parameters for SECANT should be:
A Matlab function for evaluating the function f at any given x
The initial guesses x0 and x1
The convergence tolerance TOL
The integer N0 to safeguard against infinite loops due to bugs or non-converging iterates
Use your functions to compute roots for the following three problems. For all your runs, use the
convergence tolerance
TOL = 10?15
and choose N0 large enough so that you can observe convergence or divergence. Use
λmin = 10?15
for your runs with DAMPED NEWTON.
Problem 1: Run NEWTON and SECANT to find a root of the function
f1(x) = 2x
Apply FPI to the function
the fixed point of which is the positive root of f1. Run both NEWTON and FPI with the three
initial guesses
Run SECANT with the three pairs of initial guesses
x0 = 0 and x1 = 1, x0 = 1 and x1 = 2,
Problem 2: Run NEWTON, DAMPED NEWTON, and SECANT to find a root of the function
f2(x) = arctan x
Apply FPI to the function
g2(x) = x
arctan x
the fixed points of which are the roots of f2. Run each of FPI, NEWTON, and DAMPED NEWTON
with the three initial guesses
x0 = 1, x0 = 2, x0 = 10,
for a total of 9 runs. Run SECANT with the three pairs of initial guesses
x0 = 1 and x1 = 2, x0 = 2 and x1 = 3, x0 = 10 and x1 = 11.
Problem 3: Run NEWTON, DAMPED NEWTON, and SECANT to find a root of the function
f3(x) = arctan x.
Run both NEWTON and DAMPED NEWTON with the 4 initial guesses
x0 = 1, x0 = 10, x0 = r ? 10?15, x0 = r + 10?15
,
where r denotes the value of the positive approximate root of f2 that you obtained from your run
of NEWTON with x0 = 1 in Problem 2. Run SECANT with the three pairs of initial guesses
x0 = 1 and x1 = 2, x0 = 10 and x1 = 11, x0 = r
For all your runs in Problems 1–3, print out the final value of xi+1 at termination, the corresponding
iteration index i, the total number of evaluations of the function f (or g in the case of FPI) used
in that run, and for NEWTON and DAMPED NEWTON, the total number of evaluations of the
derivative f0
. For all your runs, comment on the speed of convergence and possible divergence. If
there is divergence, provide an explanation for it.
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