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Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Jan07 667

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Problem 4

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Let , suppose and assume is nonsingular . Consider the following iteration

Problem 4a

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Derive the following error equation for

Solution 4a

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Note the following identity


The error is given by

Problem 4b

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Let be a fixed matrix. Find conditions on B that guarentee local convergence. What rate of convergence do you expect and why?

Solution 4b

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Assume is invertible, is bounded, and is Lipschitz.


This implies local superlinear convergence.

Problem 4c

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Find sufficient conditions on for the convergence to be superlinear. What choice of corresponds to the Newton method and what rate of convergence do you expect?

Solution 4c

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Super linear convergence

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as

Find condition for super linear convergence

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From part(b)



Since , if



as , we have super linear convergence i.e.


Problem 5

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Let be uniformly Lipschitz with respect to . Let be the solution to the initial value problem : . Consider the trapezoid method

.

Problem 5a

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Find a condition on the stepsize that ensures (1) can be solved uniquely for .

Solution 5a

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The implicit method can be viewed as a fix point iteration:



We want


which implies


Problem 5b

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Define a local truncation error and estimate it. Examine the additional regularity of needed for this estimate.

Solution 5b

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Re-writing (1) and replacing we have a formula for consistency of order p:



For uniform stepsize h

Expanding in Taylor Series about gives

Problem 5c

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Prove a global error estimate for (1)

Solution 5c

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Problem 6

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Consider the 2-point boundary value problem

,

with constants and . Let be a uniform partition of [0,1] with meshsize .

Problem 6a

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Use centered finite differences to discretize (2). Write the system as

and identify . Prove that A is nonsingular.

Solution 6a

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Using Taylor Expansions, we can approximate the second derivative as follows



We can eliminate two equations from the n+2 equations by substituting the initial conditions into the equations for and respectively.


We then have the system



is nonsingular since it is diagonally dominant.

Problem 6b

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Define truncation error and derive a bound for this method in terms of . State without proof an error estimate of the form

and specify the order s.

Solution 6b

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Local Truncation Error

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Bound for Local Truncation Error

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Derive Bound for Max Error

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Let , , and is the local truncation error.


Then



Subtracting the two last equations gives



Hence,


, that is the error has order 2.

Problem 6c

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Prove the following discrete monotonicity property: If is the solution corresponding to a forcing , for , and then componentwise.

Solution 6c

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Note that is a matrix and hence the discrete maximum principle applies. (See January 05 667 test for definition of matrix)


Discrete Maximum Principle

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If , then .


Specifically let , then which implies