MATH 573 ASSIGNMENT 4
1. Let T be the triangle with vertices a1 = (1, 0), a2 = (1, 1), and a3 = (0, 1). Find the
barycentric coordinates i (x), i = 1, 2, 3 of a point x T with respect to a1 , a2 , and a3 .
2a. Let T be an arbitrary triangle with vertices
MATH 573 ASSIGNMENT 5
1. The trapezoidal rule with end correction is the approximation
Z b
h
h2
f (x) dx [f (a) + 2f (a + h) + + 2f (a + [N 1]h) + f (b)] + [f 0 (a) f 0 (b)].
2
12
a
a) Derive this formula by integrating the piecewise cubic Hermite polynom
MATH 573 ASSIGNMENT 2
1a. Let I = cfw_1, 1/2, 0, 1/2, 1 and f (x) = 1/(1 + x2 ). Find the value of each of the
following at x = 3/4.
i) The polynomial P (x) of degree 4 interpolating f (x) on the set I.
ii) The piecewise linear function L(x) defined on a
MATH 573 ASSIGNMENT 3
1a. Let a = x0 < x1 < < xN = b, where xi+1 xi = h. Let i be a fixed integer satisfying
1 i N 2. Determine a piecewise polynomial function Qi (x) satisfying the following
properties:
(i) Qi (x) is a quadratic on each subinterval [xj1
MATH 573 ASSIGNMENT 1
Before beginning this assignment, copy the file Interppoly.m to the directory you will be
using when you start up Matlab.
The first two problems in this assignment study the effect of the choice of interpolation
points on the error i
Assignment 3
Math 573, Numerical Analysis I, Fall 2014
Due : 5PM Tuesday, October 14
Problem 1.(20 points.) The expressions eh , (1h4 )1 , cos(h), and 1+sin(h3 )
all have the same limit as h 0. Express each in the following form with
the best integer valu
Assignment 2
Math 573, Numerical Analysis I, Fall 2014
Due : 5PM Tuesday, September 30
Problem 1.(15 points.) A complete cubic spline S(x) of a function f (x) is
dened as follows:
S(x) =
3x + 2x2 + 2x3
a + b(x 1) + c(x 1)2 + d(x 1)3
0x1
1 x 2.
with f (0)
Assignment 1
Math 573, Numerical Analysis I, Fall 2014
Due : 5PM Tuesday, September 16
Problem 1.(10 points.) Let x0 , x1 , , xn be distinct real points and f0 , f1 , , fn
be given values. Consider the following interpolation problem: Find a function
n
ak
Exercise 3
October 20, 2014
Problem 1. Find the weights and nodes of the 2 point Gaussian formula
f (x)ex dx w0 f (x0 ) + w1 f (x1 ).
0
We note that
0
xn ex dx = n!,
0! = 1.
Problem 2. Suppose that f has a continuous second derivative on [0, 1].
Show that
Exercise 1
October 16, 2014
Problem 1. Derive the following formula for approximating the rst derivative and nd the order of accuracy:
f (x)
f (x + 2h) + 8f (x + h) 8f (x h) + f (x 2h)
.
12h
Problem 2. Establish a formula of the form
f (x) =
Af (x + 3h)
Exercise 2
October 17, 2014
Problem 1. Determine values for A, B, and C that make the formula
2
xf (x) dx Af (0) + Bf (1) + Cf (2)
0
exact for all polynomials of degree as high as possible. What is the maximum
degree?
Problem 2. We approximate the followi
Numerical Analysis I
Interpolation I
DukSoon Oh
[email protected]
Department of Mathematics
Rutgers University
September 2, 2014
Interpolation
The concept of interpolation is the selection of a function P(x) from
a given class of functions in such a
3.5
Systems of Dierential Equations
Let us consider the following system of rst order ODEs whose unknowns are
Y1 (t), Y2 (t), , Ym (t):
Y1 = f1 (t, Y1 , Y2 , , Ym )
Y2 = f2 (t, Y1 , Y2 , , Ym )
(3.5.0.11)
Ym = fm (t, Y1 , Y2 , , Ym )
with the initial cond
3.6
Solving Implicit Methods
Due to stability issues, it can be advantageous to use an implicit method. However, we must compute the numerical solution at each step by solving a nonlinear
equation.
If the trapezoidal method is used, we have the nonlinear
3.4
Absolute Stability
Since the expense of the computation increases as the step size h decreases, we
generally want to choose the time step as large as possible. We will consider a
method to estimate the size of h.
To determine whether a numerical metho
3.3.3.2 Zero Stability We need p starting values, y0 , y1 , , yp1 , before
we can apply a pstep method to the IVP y = f (t, y), y(0) = y0 . Here, y0 is given
by the initial condition, but the others y1 , , yp1 have to be computed by
other means. At any r
3.3.2
General Multistep Methods
The general pstep method has the form
p
p
aj yn+j = h
j=0
bj fn+j ,
(3.3.2.1)
j=0
where fn+j = f (tn+j , yn+j ).
We note that in order to start a calculation we need y0 , y1 , , yp1 , but
only y0 is given. We can calculate
3.2.5
Implicit RungeKutta Methods
A construction of an implicit RungeKutta method is relatively easier then that
of explicit RungeKutta methods. From Gaussian quadrature rules, we have
s
ti+1
f (, y( ) d h
ti
bj f (ti + cj h, y(ti + cj h).
(3.2.5.1)
j=
3.2.4
RungeKutta Methods
As before, we consider the IVP
y = f (t, y),
y(t0 ) = y0
(3.2.4.1)
and integrate both sides of the dierential equation from ti to ti+1 to obtain
ti+1
y(ti+1 ) = y(ti ) +
f (, y( ) d.
(3.2.4.2)
ti
Therefore, the solution to our IV
3.2.2
Taylor Series Methods
Eulers method was developed by truncating the Taylor series expansion after
just one term. We can consider higher order methods by keeping more terms in
the series. These higher order methods will be more accurate Eulers method
3
Numerical Methods for Ordinary Dierential
Equations
3.1
Preliminary
Ordinary dierential equations(ODEs) frequently occur in mathematical modes
that arise in many branches of science, engineering, and economics. Unfortunately it is seldom that these equa
2.2.8.1 Adaptive Simpsons Quadrature Rule We develop the test for
deciding whether subintervals should continue to be divided. The Simpsons
rule over [a, b] can be written as
b
I=
f (x) dx = S(a, b) + E(a, b),
(2.2.8.8)
a
where
S(a, b) =
ba
f (a) + 4f
6
a
2.2.7
Radau and Lobatto Quadrature
We have discussed two types of quadrature rules, which have the same form
n
wj f (xj ).
(2.2.7.1)
j=0
In the NewtonCotes formulas, the quadrature points are given and we were
able to nd the weights so that the result wa
2.2.5
Romberg Algorithm
Richardson extrapolation is not only used to compute more accurate approximations of derivatives, but is also used as the foundation of a numerical integration
scheme called Romberg Integration. In this scheme, the integral
b
f (x)
Orthogonal polynomials have some properties.
Lemma 2.2.6.2. Let cfw_n n 0 with degree of n = n for all n be a sequence
of orthogonal polynomials. If P (x) is a polynomial of degree m 1, then
(P, m ) = 0.
Proof. We can easily check that any monomial xk is
C.
a
t
cfw_l
r
e
c
G ri
e :, %. gi
1) .+1
i
d
'
\
G
.,r2?\t1)
S ;r r
*5APtt ,r
s
+
V
r.
t
T
t<
J

re
't
<
.5
r\,.t,E
td
(
7/.
\
e
Y
43
rf
i
1"t
"b'+
=\l
.( r
r:tJ.'
_A
a lla
? 13
r
I
A.
l,/
)_r
d^\;
r \'<
<Zl
w
tt
Jt4
\7
J =F'g
't=g,
2.2.4
Composite Rule
Like piecewise polynomial interpolation methods, we will consider a similar
method to obtain a better accuracy. We subdivide the interval [a, b] into smaller
intervals and apply quadrature formulas in each of these smaller intervals a
2.2.1.3 Simpsons Rule If f (x) is approximated by Lagrange interpolating polynomial of degree 2 and the integration is taken over [a, b] with the
interpolating polynomial as the integrand, the result is Simpsons rule.
We consider the following Lagrange in
2.1.2
Richardson Extrapolation
We will now introduce more accurate methods. We have the following formula:
f (x + h) f (x h) = 2hf (x) +
2 3 (3)
2
h f (x) + h5 f (5) (x) + . (2.1.2.1)
3!
5!
A rearrangement yields
1 2 (3)
1
1
1
[f (x + h) f (x h)]
h f (x)
A
Appendix
A.1
Big O Notation
We will introduce several standard ways of comparing two functions or two
sequences.
Let cfw_an and cfw_bn be two dierent sequences. We write
an = O(bn )
(A.1.1)
if there is a constant C such that an  Cbn  for suciently