Math 373- Sample Final Exam, summer 2021
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Math 373- Sample Final Exam, summer 2021
P1. Let I = f-1 ← -1/2← 0 ← 1/2← 1} and f (x) = 1/(1 +x2 ). Making use of the following divided difference table, find
Divided difference table
|
x - 1
-.5 0
.5
1 |
f(x) .5
.8
1
.8
.5 |
f[ , ] .6
.4
-.4 -.6 |
f[ , , ] f[ , , , ] f[ , , , , ] |
|
-.2 -.4
-.2 |
the value of each of the following at x = 3/4. Don’t have to simplify the answer.
(a) The polynomial P(x) of degree < 4 interpolating f (x) on the set I.
(b) The piecewise linear function L(x) defined on a mesh of width 1/2 interpolating f (x) on I.
(c) The piecewise quadratic function Q(x) defined on a mesh of width 1 interpolating f (x) on I.
(d) Using the error formula, find the smallest bound on the quantity x 
1] |f (x) - L(x)|
where L(x) is the approximation given in (b). As a computational check,
f (x) =\\ 
and f \\\ (x) = - 
.
P2. Derive a quadrature formula of the form Q = w0f (-1) + w1f (1/3) to approximate the integral
1
f (x)dx
-1
so that the formula is exact when f (x) is a linear polynomial.
(a) Find the weights w0 and w1 and write down the quadrature formula you use to approximate the integral. (b) Determine if the quadrature formula derived in part a) is exact when f (x) is a quadrature polynomial.
Show all steps.
P3. We are interested in constructing the Gaussian quadrature for integral
1
L[f ] = f (x)(1 -x2 )dx.
-1
(a) Show that the following polynomials
1 x x2 - 1
are orthogonal with respect to the inner product
1
(f ← g) = (1 - x2 )f (x)g(x)dx.
-1
(b) Determine H0 ← H1 and x0 ← x1 so that the quadrature formula 1-1 (1 - x2 )f (x)dx ≈ H0f (x0 ) + H1f (x1 ) is exact when f is a polynomial of as high a degree as possible.
(c) What is the highest degree polynomial for which the quadrature formula of part (b) gives the exact value of the integral.
P4. Consider the Lorenz model
x\ = 10(y - z)
y\ =x(20 - z) - y
z\ =xy - z
with x(0) = x0 , y(0) = y0 and z(0) = z0 .
(a) Write down the Euler method Yn+1 = Yn+hFn applied to the Lorenz model with uniform step-size h. (b) Performing one step of Heun’s method Yn+1 = Yn + 
(Kn(1) +Kn(2)) where
Kn(1) = F(Yn) and Kn(2) = F(Yn+hKn(1))
with h = 0.1 and x(0) = -10 ← y(0) = 10← z(0) = 30. Find K0(1) and K0(2) and give an approximation of the exact solution x(h)← y(h)← z(h).
P5. Consider the method: yn+1 = yn-1 + 
[fn+1 +4fn+ fn-1].
(a) Using appropriate Taylor series expansions, determine (for an appropriate value of r and constant Cr+1) the local truncation error in the form:
L[y(xn);h] = Cr+1hr+1y(r+1)(xn)+O(hr+2).
(b) State conditions (including definitions of all terms used) that are necessary and sufficient for this method to be convergent and use them to determine if this method is convergent.
(c) Write down the multistep method applied to solve the initial value problem
y\ = -10y and y(0) = 1
with step size h = 0.1. Determine if the method is absolute stable.
P6. We want to solve the second order initial value problem y\\ + 11y\ + 10y = 0 with initial condition y(0) =
1 ← y\(0) = 0.
(a) Express the second order ODE as a first-order system of ODEs: Y\ = F(Y ).
(b) Construct an explicit consistent second-order multistep iteration.
2022-08-18