A note on the generalized divided dierence
September 10, 2014
In the second lecture, we considered the relation between the divided dierence with distinct nodes and the derivative.
Let a = x0 < x1 < < xn = b. If f is suciently dierentiable,
f [x0 , x1 , ,

3.2.5
Implicit Runge-Kutta Methods
A construction of an implicit Runge-Kutta method is relatively easier then that
of explicit Runge-Kutta 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=

3.3.2
General Multistep Methods
The general p-step 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

3.3.3.2 Zero Stability We need p starting values, y0 , y1 , , yp1 , before
we can apply a p-step 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

3.4
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

3.6
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

3.5
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

Numerical Analysis I
Interpolation I
Duk-Soon Oh
duksoon.oh@rutgers.edu
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

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

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 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

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

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 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

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

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 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 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

3.2.4
Runge-Kutta 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

Numerical Analysis I
Interpolation I
Duk-Soon Oh
duksoon.oh@rutgers.edu
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

Numerical Analysis I
Interpolation II
Duk-Soon Oh
duksoon.oh@rutgers.edu
Department of Mathematics
Rutgers University
September 4, 2014
Newton Interpolation
Basic idea of the Newton Interpolation
Lets assume that Pn1 (x) interpolates f at x0 , x1 , , xn1

1.2.3.1
Cubic Spline
Denition Let x0 , x1 , , xn and f (x0 ), f (x1 ), , f (xn ) be given. Assume
that x0 < x1 < < xn . We seek a piecewise cubic polynomial S(x)
S0 (x)
S (x)
1
S(x) =
Sn1 (x)
x0 x x1
x1 x x2
.
xn1 x xn
(1.2.3.1)
The cubic polynomial Sj (x

For complete cubic splines, m0 and mn should be specied and S(x) must
satisfy the following conditions:
S (x0 ) = m0 ,
S (xn ) = mn .
(1.2.3.26)
We recall that
Si1 (x) = Mi1
and
Si (x) = Mi
(xi x)2
(x xi1 )2
+ Mi
+ Ci1
2hi1
2hi1
(1.2.3.27)
(xi+1 x)2
(x xi

Let us assume that a triangle T has three distinct vertices at (xi , yi ), (xj , yj ),
and (xk , yk ). We will nd a linear function i (x, y) on T with the following
properties:
i (xi , yi ) = 1
i (xj , yj ) = 0
i (xk , yk ) = 0.
We note that i (x, y) h

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2
2.1
2.1.1
Numerical Dierentiation and Integration
Numerical Dierentiation
Dierentiation via Taylor Series
We consider the following Taylor series:
f (x + h) = f (x) + hf (x) +
h2
f (x) + = f (x) + hf (x) + O(h2 ). (2.1.1.1)
2
We may rewrite the formula

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 | C|bn | for suciently

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