Adaptive methods:
Error estimation: When we have an approximation method of a known order, say I I h e h
where I is the exact value, I h is the approximation, e h is the error, and we know
e h Ch k O h k 1 then we can use approximations I h and I h/2 , th
Math 420 HW3
1. If x represents a vector with n components then we have for constants C, c independent of
x,
cx
x 2 C x . Find, in terms of n, the best possible constants C, c : the smallest
possible such C and largest possible such c. Explain.
1/2
1/2
(E
Math 420 HW3
1. If x represents a vector with n components then we have for constants C, c independent of
x,
cx
x 2 C x . Find, in terms of n, the best possible constants C, c : the smallest
possible such C and largest possible such c. Explain.
1/2
1/2
(E
Math 420 Hw5
1. Develop an approximation formula f 0
directed below:
a) Interpolate f at x
w1f 1
w2f 2
w 3 f 4 in several ways as
1, 2, 4 with a polynomial p 2 x in Lagrange form, and calculate f 0
1 w 1 f x 0 h w 2 f x 0 2h w 3 f x 0 4h
b) Expand the sca
Homework Due Monday March 18
1. Calculate by hand the divided difference table corresponding to the data
x201
3
and then write the polynomial in Newton form that interpolates the data.
y 5 1 1 3
(Use fractions, not decimals)
2. Write out by hand the lin
Math 420 HW6
1. Derive quadrature formulas
i)
ii)
h
0
h
0
f x dx
h w1f h
w2f 0
f x dx
h w1f 0
w2f h
w3f h
w 3 f 2h
1
a) Use polynomial interpolation as applied to the unscaled formula for f x dx and scale the
0
result
1
b) Use exactness for f x
1, x, x 2
Homework Due Monday March 18
1. Calculate by hand the divided difference table corresponding to the data
x201
3
and then write the polynomial in Newton form that interpolates the data.
y 5 1 1 3
(Use fractions, not decimals)
x
f[x(i)]
2
5
2
0
1
4
2
1

Math 420 Hw5
1. Develop an approximation formula f 0
w 1 f 1 w 2 f 2 w 3 f 4 in several ways as
directed below:
a) Interpolate f at x 1, 2, 4 with a polynomial p 2 x in Lagrange form, and calculate f 0
Our function p 2 x that interpolates the values f x i
Numerical Analysis
Spring 2005
Prof. Diamond
Applications of the error formula for polynomial interpolation:
I. Interpolation of fx cos=x on 0, 1 :
Consider the interpolation error in interpolating fx cos=x on the interval 0, 1 . Well
use equally spaced i
Supplementary notes on interpolation
The interpolation problem:
Given a set of ordered pairs (or points in the plane) x 0 , y 0 , x 1 , y 1 , , . . , x n , y n ,
find a function x from a given family of functions such that x i y i , i 0, . . , n.
Graphic
1
Lecture 7 (Tue, Sept. 11). Fixedpoint iteration and
Newtons method
7.1 Convergence rate of the xedpoint iteration
If p is a xed point of g (x), g (x) is continuous in a neighborhood of p, and g (p) < 1, then
cfw_pn converges to p for p0 close enoug
Fancy derivation of error in Newtons method
We previously obtained an estimate of the error behavior in Newtons method, based on
fx
analysis of the iteration x n 1 g x n where g x
x
fx
Here is "slicker" approach:
First, the definition is x n 1 is that it
Numerical solution of ODEs
y
f t, y , y t 0
y0
While there are methods that try to obtain functional forms that approximate the solutions,
we will look at methods that attempt to approximate the values y t i at a set of values
t i . We will let y i denote
Notes on integration formulas:
Suppose we have an integration rule
b
a
n
w i f t i that is exact for all polynomials up to degree k (called the precision, or
f x dx
i1
the degree, of the method). In any given example, we might have k equal to n 1 (the
"no
Math 420 Hw2
1
11
ln 1
ln 1 1
2
2
2
22
1 1 k the error is
If we stop at the term
k2
k1
1
1
1
1 k 2 .
12 n
n
k12
k22
1. a) ln 2
nk1
2
1
3
1
2
1
k
1
3
2
.
1
n
nk1
k
2
11
k1
1/2
1
k
1
2
k
so that the error is roughly the size of (and no bigger than) the last
This is an example of how the condition number relates to the error in the solution produced
by an error in the data. We construct a "worstcase" scenario for a random 100x100 matrix
to illustrate the potential hazards involved in solving large linear sys
Condition number of a matrix Assume A is square and consider problems Ax b. The condition number arises when
considering either of the following two problems, which are essentially equivalent:
As before, we let x denote the exact solution, and x an approx
Math 420 Exam topics and examples
1) Taylor series, Taylor polynomials and Taylor remainder
log x, sin x, cos x, exp x , 1 , 1 x 1/2 , 1 x p
1x
Ex: estimate the error from the Taylor remainder: 1 x
1 1 x if 0 x
2
general x 0
2
n
Ex: If a is a constant, sh
Derivative approximation formulas are usually formulated in terms of a scaling parameter h.
The simplest one is from first semester calculus:
f x0 h f x0
f x0
referred to as the two point forward difference formula for f x 0
h
W e also have the approximat
Math 420 Exam 3
1. The central difference formula for the second derivative is
1 f x 0 h 2f x 0
f x0
f x0 h
h2
a) Expand in powers of h to determine the leading term in the error expansion.
b) This formula is exact for polynomials f x up to what degree (c
Math 420 Review topics for exam 2
vector norms: 1, 2,
the matrix norm induced by a vector norm  what is A and what does it mean? What
about Ax
A x , AB
A B  why are they true?
Why is A equal to the maximum (absolute) row sum of A
the condition number A
Math 420 Exam 3 review topics
Derivative formulas:
1) Given a formula, expand in powers of h and determine what it approximates and the
leading error term.
2) Derive a formula (and leading error term) at a specified set of point through the method
of Tayl
Math 420 Exam 3
1. (10) The central difference formula for the second derivative is
1 f x 0 h 2f x 0
f x0
f x0 h
h2
a) Expand in powers of h to determine the leading term in the error expansion.
f x0
h
2f x 0
f x0
h
f x0
2 f x0
f x0
1 fx
0
h2
h
2f x 0
f x
Math 420 Exam topics and examples
1) Taylor series, Taylor polynomials and Taylor remainder
log x, sin x, cos x, exp x , 1 , 1 x 1/2 , 1 x p
1x
Ex: estimate the error from the Taylor remainder: 1 x
1
a general x 0
f
1 x if 0
2
x
0. 1 , or for
n1
x a n 1 w
Extrapolation:
If we have an approximation in terms of a scale parameter h, and we know the order of the
leading term in the error, then we can combine the approximation obtained at two different
scales so as to obtain an approximation with a higher order
Review topics:
Note: You may bring one piece of paper to the final with whatever notes you want.
Roundoff error, absolute error, relative error and effect of arithmetic operations on these
Taylors theorem and Taylor expansions
Fixed points  where are the