MCS 471 Homework 1: Nonlinear Equations
(Revised 2/07/96)
NOT Due: Practice Problems ONLY! (revised 2/07/96)
Bckground Reading: Gerald and Wheatley, Chapter 1
Topics
- Bisection Method
- Secant Method
- Newton's Method
- Fixed Point Interation
- Muller's Method
- Golden Section Search (See Lecture Notes, Not in Text)
Homework Problems:
In problems 1-7, use "EXAM PRECISION": Chop to 4
significant (4C) digits only when you write an intermediate or final
answer down and continue calculations with those numbers recorded.
{Change: Problems 1-4 were changed with 'F(x)=exp(x)-3.5/x' replacing
'F(x)=exp(x)-2.5/x' so that there is a zero in [1,2].
- Using the method of BISECTION starting with A(1)=1. and B(1)=2.,
find the root of
F(X)=EXP(X)-3.5/X
Record your answer in a table of
K,A(K),B(K),F(A(K)),F(B(K))
for each iteration on (A(K),B(K)) for K=1 to 3.
Compare your answer to that using fsolve of Maple.
- Find the root of
F(X)=EXP(X)-3.5/X
using the SECANT METHOD for 2 iterations beyond the starting guesses,
X(1)=1. and X(2)=2. Record your answers in a table of
K,X(K),X(K-1),F(X(K)),F(X(K-1))
for each iteration K.
Compare your answer to the Bisection and Maple
answers from the first question.
- Find the root of
F(X)=EXP(X)-3.5/X
on [1.,2.] using NEWTON'S METHOD until ABS(X(K)-X(K-1))<.5E-1.
Record Results in table of
K,X(K),F(X(K)).
Use X(1)=1.5 to start.
Compare your answer to the Bisection, Secant and Maple
answers from the first and second questions.
- Find the root of
F(X)=EXP(X)-3.5/X
on [1.,2.] using MULLER'S METHOD until ABS(X(K)-X(K-1))<.5E-1.
Record Results in table of
K,X(K),F(X(K)).
Use {1,2,1.5} as starting values.
Compare your answer to the Bisection, Secant, Newton and Maple
answers from the first through third questions.
- Numerically solve
F(X)=LN(X)-1/X=0
by forming a convergent, fixed point iteration, other than Newton's,
starting from X(1)=EXP(1). Record your answers in a table of
K, X(K), for K= 1 to 3.
Compare your answer to that using fsolve of Maple.
Use the plot function of Maple to plot the problem function
G(X).
- Find the minimum of
G(X)=EXP(X)+7.8/X
on (1,2) by the method of
GOLDEN SECTION SEARCH for K=1,2,3 iterations. Display your answer in a
table of
K, AK, BK, XK, UK, GXK, GUX.
Use Maple's plot to plot the function G(X) and
compare your problem answer to that using minimize of Maple.
- Find the maximum and its location for
G(X)=X*COS(X)
on (0.4,1.4) by
the method of GOLDEN SECTION SEARCH. Summarize your results by a
tabulation of
K, AK, BK, XK, UK, GXK, GUK
for K=1 to 3.
{WARNING!: Ignore the Maple part of this exercise since "maximize"
obviously does not work for this simple trigonometric function. "Maximize"
seems to work primarily for polynomial functions over algebraic fields
(revised 2/07/96).
However, it may work if
you approximate "X*cos(x)" by the first few terms of its Taylor series.
Use Maple's plot to plot the function G(X) and
compare your problem answer to that using maximize of Maple.}
- Using Maple, get all the roots, including double and triple roots,
of the polynomial
x^5-11x^4+46x^3-90x^2+81x-27.
Also plot the polynomial using the plot function of Maple, on
[0,4].
Web Source: http://www.math.uic.edu/~hanson/M471/mcs471hw1.html
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