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Math 4513
Solutions to Homework Set 1
1.1. Given that
dn
dxn
ln jxj = ,1n,1 n , 1!x,n+1
a Use the Taylor Theorem with Integral Remainder to nd the magnitude of the error term R1001:99
when one approximates ln 1:99 using the rst 101 terms of the Taylor exp
MATH 4513 : HOMEWORK 5
1. Solve the following linear systems twice. First use Gaussian elimination and give the factorization A = LU. Second, use Gaussian elimination with scaled pivoting and determine the factorization of the form PA = LU. a
0 ,1 1 ,4 1
MATH 4513 : HOMEWORK 8
1. Write down the Richardson extrapolation for the derivative f x that is accurate to order 8 in h.
0
2. Suppose
Zb
a
f xdx
is calculated numerically by interpolating the function f x at the points
i + 1
a+b b,a
; i = 0; 1; 2; : :
1. Determine the order of the following multi-step methods.
a
x
n , xn,2 = 2hfn,1
b
x
n , n,2 =
x
c
x
n , xn,1 = h
h
7
, 2 n,2 + 1 n,3
3 n,1 3
3
f
f
f
3 + 19
5
1
n
n,1 ,
n,2 +
8
24
24
24 n,3
f
f
f
f
2. Consider the following intial value problem
=
1 =
dx
MATH 4513 : HOMEWORK 7
1. Write a Maple program that nds the optimal set of + 1 nodes on a speci ed interval in such a way that it takes the numbers and as input parameters.
n n, a, b n
a; b
. Write it
n
2. Write a Maple program that nds the Newton form o
MATH 4513 : HOMEWORK 6
1. Find the Newton and Lagrange forms for the interpolation polynomial corresponding to the following
sets of data.
a
=0
=1
2=2
3=3
= ,1
= ,2
2 = ,1
3 = ,4
x0
y0
x1
y1
x
y
x
y
b
=1
=2
2=0
3=3
=3
=2
2 = ,4
3=5
x0
y0
x1
y1
x
y
x
y
2.
MATH 4513 : HOMEWORK 4
1. If
01 0 0 01
B3 4 1 0C
L =B 2 1 0 0 C
A
@
;
4 5 6 1
Write a program that nds the solution of
011
B3C
b=B 2 C
@ A
4
Lx = b
:
2. If
01 1 2 11
B0 0 2 1C
U =B 0 2 1 2 C
@
A
;
0 0 0 1
Write a program that nds the solution of
011
B3C
b
MATH 4513 : HOMEWORK 3
1. Show that
e
n+1
00 2jf 0rj jf rj
expx
ee
n n,1
=
Ce e
n n,1
2. Use the secant method to nd a solution of starting with
x0
= 1:5;
x1
= 1:4.
2 , 2 = 3lnx
1
MATH 4513 : HOMEWORK 2
1. Let n n denote the sucessive intervals that arise from applying the bisection method to nd a root of a continuous function x. Let n = 1 an + n, = limn!1 n , and n = , n. 2
a ;b f c b r c e r c
a Show that jen j 2,n jb , b Show th
MATH 4513 : HOMEWORK 9
1. Use the Taylor series method to calculate a solution to
dx
dt
=
2
x t
;
x
0 = 1
that's accurate to order 4 .
t
2. Use a ve step Euler method to calculate the solution of
dx
dt
on the interval 0 1
;
=
2
x t
;
x
0 = 1
:
3. Use a ve
Math 4513
Solutions to Homework 6
1. Find the Newton and Lagrange forms for the interpolation polynomial corresponding to the following
sets of data.
a
x0
x1
x2
x3
=0
=1
=2
=3
y0
y1
y2
y3
= ,1
= ,2
= ,1
= ,4
The Newton form of the interpolation polynomial
Math 4513
Solutions to Homework 7
1. Write a Maple program that nds the optimal set of + 1 nodes on a speci ed interval
in such a way that it takes the numbers , , and as input parameters.
n
n
a
a; b
. Write it
b
values for n , a, b must already be define
MATH 4513 : HOMEWORK SOLUTIONS 5
1. Solve the following linear systems twice. First use Gaussian elimination and give the factorization
A = LU. Second, use Gaussian elimination with scaled pivoting and determine the factorization of the
form PA = LU.
a
0
Math 4513
Solutions to Homework 10
1. Determine the order of the following multi-step methods.
a
xn , xn,2 = 2hfn,1
This equation corresponds to a two step method with coe cients
a2 = 1 ; a1 = 0 ; a0 = ,1
b2 = 0 ; b1 = 2 ; b0 = 2
We have
d0 =
d1 =
d2 =
d3
LECTURE 1
Variations on Taylor's Formula
Numerical methods are ipso facto approximate methods. This being the case, it will be important throughout this course to determine the accuracy of numerical results. We shall begin by reviewing the analytic
method
LECTURE 2
Sequences and Limits
1. De nition of Sequences and Limits
The most common situation In Numerical Analysis, is when one is faced with a problem in which an exact
number can never be found; yet by successive approximations a highly accurate determ
LECTURE 4
Floating Point Error Analysis
1. Absolute and Relative Errors
2. Relative Error Analysis
We now state two theorems regarding the propagation of round o errors for sums and products.
Theorem 4.1. Let x0 ; x1; : : : ; xn be positive machine number
LECTURE 3
Computer Arithmetic
Anyone who has ddled with a calculator long enough knows that it's relatively easy to trick it into producing
ridiculous answers. Indeed, in spite of the possibility of producing answers with a huge number of decimal
places,
Math 4513
Solutions to Homework 8
1. Write down the Richardson extrapolation for the derivative f 0 x that is accurate to order 8 in h.
In lecture we derived
where
f 0 x = 0 h + Oh2
= 1 h + O,h4
= 2 h + O h6
,
= 3 h + O h8
=
f x + h , f x , h
2
and the
MATH 4513 : HOMEWORK 3
1. Show that
n+1
e
00
2j j 0 jj
f
r
f
r
n n,1
ee
=
n n,1
Ce e
Let be the actual root of = 0, let n be the approximate value for obtained by carrying
out iterations of the secant method, and let n be the corresponding error:
n = n,
MATH 4513 : HOMEWORK 2
1. Let an ; bn denote the sucessive intervals that arise from applying the bisection method to nd a root of
a continuous function f x. Let cn = 1 an + bn, r = limn!1 cn , and en = r , cn.
2
a Show that jen j 2,n jb , aj.
At the star
MATH 4513 : HOMEWORK 4
1. If
01 0 0 01
B3 4 1 0C
L =B 2 1 0 0 C
@
A
;
4561
Write a program that nds the solution of
011
B3C
b=B 2 C
@A
4
Lx = b
:
n := 4;
L := array1.n,1.n;
b := array1.n;
x := array1.n;
L :=
1,0,0,0 , 2,1,0,0 , 3,4,1,0 , 4,5,6,1 ;
b := 1,
MATH 4513 : HOMEWORK 1
1.1. Given that
n
ln jxj
dxn
d
x=1
= ,1n,1 , 1!
n
a Use the Taylor Theorem with Integral Remainder to nd the magnitude of the error term 1001 99
when one approximates ln 1 99 using the rst 101 terms of the Taylor expansion about 1 o