Assignment 3
Problem 1
a. The function L1 (x, y ) satises
L1 (1/2, 0) = 1, L1 (0, 1/2) = 0, L1 (1/2, 1/2) = 0
If L1 (x, y ) = ax + by = c, this implies
1
2a + c = 1
1
b+c=0
2
1
a + 1b + c = 0
2
2
whic
Assignment 3
Problem 1
a. Since Qi (x) and Qi (x) are continuous, Qi (xi1 ) = Qi (xi1 ) = 0 and Qi (xi+2 ) =
Qi (xi+2 ) = 0.
First nd a function Q with the required properties in the interval [1, 2],
Linear Algebra and Numerical Analysis: Matlab
Martin Mensik
1
Martin Mensik
Linear Algebra and Numerical Analysis : Matlab
Basic routines
Write a script le (not a function) that will do following task
MATLAB FOR NUMERICAL ANALYSIS
Home Exercises:
(Submit the printed figures and necessary commands for the following two exercises next week.)
Question1. For each of the following functions, find an int
Numerical Analysis Lab Note #1
Matlab Basic
=
Variables
Like most other programming languages, the MATLAB language provides
mathematical expressions, but MATLAB does not require any type
declarations
Numerical Analysis Lab Note #2
Matlab Basic
Matrix, Vector, Function, and Script M-File
=
Matrices and Vectors
In Matlab, all variables can be viewed as matrices.
You can use "[ ]" to define a matrix.
MATLAB has many tools that make this package well suited for numerical computations. This
tutorial deals with the rootfinding, interpolation, numerical differentiation and integration and
nume
MATLAB Commands and Functions
Dr. Brian Vick
Mechanical Engineering Department
Virginia Tech
General Purpose Commands
Operators and Special Characters / 3
Commands for Managing a Session / 3
Special V
NUMERICAL ANALYSIS PROGRAMS IN MATLAB
This README le gives instructions for the Matlab programs on the disk. The programs are designed
to run on a minimally congured computer. Minimal hard disk space
Math 373 Assignment 4
Problem 1 Let V2 denote the space of continuous, piecewise quadratic functions defined
on a partition P : a = x0 < x1 < < xN = b , where xi = a + ih and h = (b a)/N .
And let xj1
Assignment 1
Problem 4
a. Since P3 (0) = P3 (1/3) = P3 (2/3) = 0, we must have
2
1
P3 (x) = Ax(x )(x )
3
3
for some constant A. Now evaluating P3 (x) at x = 1 and using the condition P3 (1) = 1,
we ge
MATH 573 ASSIGNMENT 6
1. Determine the two point Gaussian quadrature formula and error term for approximating
1
f (x) dx.
0
2. In this problem, we develop a quadrature formula of the form
b
w(x)f (x)
1. 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
MATH 573 LECTURE NOTES
57
13.4. Estimation of local error. In practice, we not only want to produce an approximation to the solution at each step of the algorithm, we also want to produce an estimate
0.1. Linear multistep methods. The general linear (p + 1) step method has the form
p
yn+1 =
p
ai yni + h
bi fni ,
i=1
i=0
where fni = f (xni , yni ) and the ai and bi are constants.
Remarks: Any of th
68
MATH 573 LECTURE NOTES
13.6. Stability of linear multistep methods.
Denition: 1st and 2nd characteristic polynomial of a multistep method:
p
(z ) = z
p+1
p
ai z
pi
,
bi z pi .
(z ) =
i=0
i=1
The l
MATH 573 LECTURE NOTES
71
13.7. Strong, weak, absolute and relative stability. To formalize the stability problem
discussed above, we now dene several concepts of stability that seek to dierentiate be
74
MATH 573 LECTURE NOTES
13.8. Predictor-corrector methods. We consider the Adams methods, obtained from the
formula
xn+1
xn+1
f (x, y (x) dx
y (x) dx =
y (xn+1 y (xn ) =
xn
xn
by replacing f by an i
MATH 573 ASSIGNMENT 1
Before beginning this assignment, copy the le Interppoly.m from the RESOURCES folder
to the directory you will be using when you start up Matlab.
The rst three problems in this a
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
MATH 573 ASSIGNMENT 3
1a. Let a = x0 < x1 < < xN = b, where xi+1 xi = h. Let i be a xed integer satisfying
1 i N 2. Determine a piecewise polynomial function Qi (x) satisfying the following
properties
MATH 573 ASSIGNMENT 4
1a. Let T be the triangle with vertices a1 = (1, 0), a2 = (0, 1), a3 = (0, 0). Find linear
polynomials L1 (x, y ), L2 (x, y ), L3 (x, y ) in x and y such that any linear polynomi
MATH 573 ASSIGNMENT 5
b
1. One form of the error for the approximation of a f (x) dx by the composite rectangle rule
h N 1 f (a + ih) (where h = (b a)/N ) is given by
i=0
N 1
b
f (x) dx h
a
f (a + ih)
1. Cubic spline approximation
1.1. 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 additiona