ESS 305002 Numerical Analysis
Midterm exam (130pts)
23 Nov 2006
1.
(a) [10pts] The roots of x 2  83.4 x  1  0 are approximately x1  0.01199213 and
x2  83.38800785 . Suppose now that you work under four-digit rounding arithmetic.
Calculate x1 using the following two different formulas: x1 
x1 
 b  b 2  4ac
and
2a
 2c
. Compute the relative error obtained by the two formulas.
b  b 2  4ac
Explain why the second formula gives a more accurate answer than the first one.
(b) [10pts] Let P( x)  a n x n  a n 1 x n 1    a1 x  a0 be a polynomial. Construct an
algorithm to evaluate P( x0 ) at given x  x0 , using nested multiplication.
2. Suppose that f  C 2 [a, b] . Let p0  [a, b] be an approximation to the solution p
of f ( x)  0 such that f ' ( p0 )  0 and p  p0 is small.
(a) [10pts] Newton-Raphson method generates an sequence
pn n0 , which starts
with the initial approximation p0 and tends toward the solution p as n   . Please
state what the Newton-Raphson method is and how to generate pn.
(b) [10pts] Secant method is a modification of the Newton-Raphson method. Please
state what this method is and how it works. Please point out the weak points of the
Newton-Raphson method which people want to circumvent by using the secant
method.
(c) [10pts] Mueller’s method is a generalization of the secant method. Please state
what it is and how it generates a sequence
pn n0
converging to the solution p.
(d) [10pts] Write down an algorithm for the secant method, given the two initial
approximations p0, p1, the error tolerance TOL, and the maximum number of iteration
N0 .
3.
(a) [10pts] P(x) is the Lagrange interpolating polynomial to approximate the function
f(x) passing through the (n+1) distinct points ( x j , f ( x j )) for j  0,1,, n . Please
write down P(x).
(b) [10pts] Denote Qi , j ( x) , for 0  j  i , the Lagrange interpolating polynomial of
degree j on (j+1) numbers xi  j , xi  j 1 ,  , xi . Show that
Qi , j ( x) 
( x  xi  j )Qi. j 1 ( x)  ( x  xi )Qi 1. j 1 ( x)
xi  xi  j
(c) [10pts] What is Neville’s iterated method and how does it work? Use Neville’s
method to evaluate the Lagrange interpolating polynomial P(x) at the number x=1.7
for the function f(x) passing through the 4 distinct points in the table.
xi
f(xi)
1.0 0.7652
1.3 0.6201
1.6 0.4554
1.9 0.2818
4.
(a) [10pts] Define a function f defined on [a,b] and a set of nodes
a  x0  x1    xn  b . The cubic spline interpolation is a piecewise-polynomial
approximation using cubic polynomials between each successive pair of nodes. Please
state, respectively, the conditions which a cubic spline interpolant S for f should
satisfy under free boundary and under clamped boundary.
(b) [10pts] Denote S j ( x)  a j  b j ( x  x j )  c j ( x  x j ) 2  d j ( x  x j ) 3 a cubic
polynomial on the interval [ x j , x j 1 ] and h j  x j 1  x j , for j  0,1,, n  1 . It can
be derived that under clamped boundary, c j is the solution of the linear system Ax=b,
where
Show that (1) a j  f ( x j ) ,
bj 
(2) d j  (c j 1  c j ) / 3h j ,
(3)
1
(a j 1  a j )  c j h j  d j h 2j
hj
(c) [10pts] Construct the clamped cubic spline for the data
x
f(x)
-1
0.8620
-0.5 0.9580
0
1.099
0.5
1.294
and f’(-1)=0.1554 and f ’(0.5)=0.4519
5. [10pts] Obtain factorizations of the form A  PLU for the matrix
1  2 3 0 


1  2 3 1 
A
by means of Gaussian elimination, where P is a permutation
1  2 2  2


2 8 3 1


matrix, L is a lower-triangular matrix with 1s on the diagonal, and U is a
upper-triangular matrix. Determine det A =?