64
MATH 573 LECTURE NOTES
13.5.3. Convergence of multistep methods. Denition: The linear multistep method dened
by the formula
p
(13.1)
yn+1 =
p
ai yni + h
i=0
bi fni ,
i=1
is said to be convergent, if for all initial value problems y = f (x, y ), y (a) =

6
MATH 573 LECTURE NOTES
2. Polynomial Interpolation
2.1. Interpolation error. We now turn to an analysis of the error f () Pn (), for x =
x
x
x0 , . . . xn . For the moment, consider x xed, and let Pn+1 denote the polynomial of degree
n +1 interpolating

10
MATH 573 LECTURE NOTES
2.3. Interpolation of moments. In some applications, it is useful to interpolate quantities
other than function or derivative values. One quantity that arises frequently is the moments
b
of a function f over an interval [a, b], i

MATH 573 LECTURE NOTES
13
3.1. Piecewise linear approximation. Consider in more detail the case of continuous,
piecewise linear approximation. Dene the continuous, piecewise linear interpolant of a
function f as the continuous, piecewise linear function Q

16
MATH 573 LECTURE NOTES
4. Cubic spline approximation
4.1. Cubic spline interpolation. We consider the problem of nding a C 2 piecewise cubic
function S (x) that satises S (xi ) = f (xi ), i = 0, . . . , n plus two additional conditions. These
are usual

MATH 573 LECTURE NOTES
19
5. The Finite Fourier Transform
We consider the approximation of a periodic function f with period 2 , i.e., f (t + 2 ) =
f (t). Note that a function with a more general period can be reduced to this case in the
following simple

MATH 573 LECTURE NOTES
23
6. Piecewise polynomial approximation in two dimensions
We consider the approximation of a function u(x, y ), where u is dened on a convex
polygon . For each 0 < h < 1, we let Th be a triangulation of into triangles Ti with the
f

MATH 573 LECTURE NOTES
27
7. Approximation of Derivatives
Basic idea: replace the function by its interpolating polynomial and use the derivative of
the interpolating polynomial as an approximation to the derivative of the function.
7.1. Numerical Dierent

30
MATH 573 LECTURE NOTES
8. Approximation of Integrals
Basic idea: replace the function by its interpolating polynomial and use the integral of the
interpolating polynomial as an approximation to the integral of the function.
8.1. Basic Numerical Integra

34
MATH 573 LECTURE NOTES
9. Approximation of Integrals Continued
9.1. Iterative Approaches to the Approximation of Integrals. One of the simplest
iterative procedures to use is interval doubling. For example, using the composite trapezoidal
rule on N sub

MATH 573 LECTURE NOTES
37
10. Gaussian Quadrature
10.1. Quadrature formulas with given abscissas. We have previously seen that one
way of obtaining quadrature formulas of the form
n
b
f (x) dx =
a
Hj f (xj ) + E
j =0
in the case when the xj are specied is

MATH 573 LECTURE NOTES
41
10.4. Construction of Gaussian quadrature formulas. Using these results, we now
return to the problem of nding abscissas x0 , , xn and weights H0 , , Hn so that
n
b
Hj P (xj )
w(x)P (x) dx =
a
j =0
for all polynomials P (x) of de

44
MATH 573 LECTURE NOTES
11. Adaptive Quadrature
Previously, we considered a simple interative algorithm based on interval doubling for
computing an approximation to the value of an integral. In such a method, based on the
composite trapezoidal rule, for

50
MATH 573 LECTURE NOTES
13. Numerical solution of Ordinary Differential Equations: Background
Consider the initial value problem (IVP) for a rst order ordinary dierential equation:
dy/dx = f (x, y ),
y (x0 ) = y0 .
The following theorem gives sucient co

54
MATH 573 LECTURE NOTES
13.3. Runge-Kutta methods. We now consider a class of methods, called Runge-Kutta
methods, that achieve the same accuracy as Taylor series methods, without calculating
derivatives of f . The basic idea is to use a linear combinat

60
MATH 573 LECTURE NOTES
13.5. Linear multistep methods. The general linear (p + 1) step method has the form
p
yn+1 =
p
ai yni + h
i=0
bi fni ,
i=1
where fni = f (xni , yni ) and the ai and bi are constants.
Remarks: Any of the ai s and bi s may be zero,

2
MATH 573 LECTURE NOTES
1. Polynomial Interpolation
1.1. Weierstrass Approximation Theorem. If f (x) is continuous on a nite interval
[a, b], then given > 0, there exists n depending on and a polynomial Pn (x) of degree n
such that |f (x) Pn (x)| for all