Math 417, Quiz # 1
Name & ID number:
Texas A & M University
Fall 2011
1 (a) [15 points] Use 2 iterations of the bisection method to calculate the
solution of x3 7x2 + 14 6 = 0 on [0, 1].
1 (b) [5 points] How many steps are needed to get accuracy 103 ?
Math 417, Homework 7
1. Consider
410
A = 1 4 1 ,
014
(a) Find B
ods.
,
5
b= 6
5
(B ), where B is the iteration matrix for the Jacobi and Gauss-Seidel meth-
(b) Use the theorem given in class to nd the optimal SOR parameter (you should check that
A satise
Math 417, Homework 6
1. Use the nite dierence scheme to solve the 2-point boundary value problem
+ = 2x + 1,
0 < x < 1,
(0) = (1) = 0.
Use mesh size h = 2n , n = 1, . . . , 8. Take = 103 . Solve Ah u = f using LU factorization
for a tridiagonal system as
Math 417, Homework 5
1. Consider the system of linear equations
x1 + 2 x2 + 3 x3 = 6
x1 + x2 + 2x3 = 2
4x1 + x3 = 5
(a) Write the system in the form (A|b). Solve by reduction to triangular form and back
substitution.
(1) (2) (3)
(b) Show that detA = a11 a
Math 417, Homework 4
1. (a) Derive the following rule for estimating f
f (x)
1
(f (x + 2h) 2f (x + h) + 2f (x h) f (x 2h) ,
2h3
and the corresponding error term.
(b) Use the centered dierence approximation D0 f to estimate f (x) for f (x) =
with h = 0.01
Math 417, Homework 3
1. Let x0 , . . . , xn be distinct points. In Hermite interpolation, the function hj (x) was dened to
be the polynomial of degree 2n + 1 satisfying
hj (xi ) = 0,
hj (xi ) = 0, i = j,
(a) Verify that hj (x) = (x xj ) j (x)2 , where
hj
Math 417, Quiz # 5
Name & ID number:
Texas A & M University
Fall 2011
1.[10 points] Use the Linear Finite -Dierence method to approximate the
solution of the boundary value problem
y = 4(y x), 0 x 1, y (0) = 0, y (1) = 2,
with h = 1/2.
2.[15 points] Let x
Math 417, Quiz # 4
Name & ID number:
Texas A & M University
Fall 2011
1.[30 points] Consider the following integral.
2
2
xe2x dx
1
a ) Construct a 3 row Romberg integration table.
b ) What is the relative error of the approximation in (a) using R22 ?
c )
Math 417, Quiz # 3
Name & ID number:
Texas A & M University
Fall 2011
1.[10 points] Determine the order of accuracy of f (x)
2.[10 points] Use Simpsonss rule to approximate
1.5
1
f (x + h) f (x h)
2h
x2 ln xdx.
3.[10 points] Find a bound for the error in
Math 417, Quiz # 6
Name & ID number:
Texas A & M University
Fall 2011
1. Consider the system
3x y = 5
2x + 3y = 4
a ) [3 points] Does the Jacobi iterative method converge?
b ) [10 points] Find the rst three iterations of the Gauss-Siedel method
using X (0
Math 417, Homework 2
1. Let f (x) =
1
5+x .
Take x0 = 1, x1 = 2, x2 = 3, x3 = 4,
(a) Find the Lagrange form, the Newton form and the standard form of the interpolating
polynomial L3 . Check your answer by veryfying that L3 correctly interpolates f at the
Math 417, Homework 1
1. Consider f (x) = x3 2.
(a) Show that f (x) has a root in the interval [1, 2].
(b) Compute an approximation to the root by taking 4 steps of the bisection method.
(c) Repeat, using xed point iteration with g1 (x) = x f (x)/3 and g2