Queen Mary and Westfield College

UNIVERSITY OF LONDON

B.Sc. EXAMINATION

MAS/104 Computational Mathematics II

Summer 1999

The duration of this examination is 2 hours.

This paper has two Sections and you should attempt both Sections. Please read carefully the instructions given at the beginning of each Section.

SECTION A: This section carries 40 marks and each question carries 4 marks. You should attempt ALL 10 questions.

SECTION B: This section carries 60 marks and each question carries 20 marks. You may attempt all 4 questions but only marks for the best 3 questions will be counted.

Calculators may be used in this examination but any programming, graph plotting or algebraic facility may not be used. Please state on your answer book the name and type of machine used.

SECTION A

This section carries 40 marks and each question carries 4 marks. You should attempt ALL 10 questions. Do NOT begin each answer in this section on a fresh page. Write the number of the question in the left margin.

Write the Maple statements necessary to perform each of the following computations:

1. Generate the sequence consisting of all the integers between 1 and 100 inclusive in reverse order, i.e. 100,99,98,...,3,2,1 ( without writing every element explicitly).

2. The variable L evaluates to a list containing the element a . Delete this element from the list, i.e. construct a copy of the list L without the element a .

3. Construct from an arbitrary list L the list with the same elements in reverse order.

4. The variable S evaluates to a set of sets. Construct the union (i.e. the set of all the elements) of all the sets in S .

5. Find the number of elements in the largest set in a set S of sets.

6. Define the mapping E such that

.

The definition must allow n to be symbolic, in which case it will attempt to evaluate the sum symbolically and remain as a symbolic summation if it fails.

7. Define the mapping f such that

and plot it on the domain [-1, 1].

8. Define a mapping dfseq that takes as arguments ( y , x , n ) an expression y that depends on a variable x and a positive integer n , and returns a sequence of y followed by its first n derivatives with respect to x , e.g.

9. Evaluate exactly, and evaluate numerically, accurate to 15 significant figures, without attempting to evaluate it exactly.

10. Plot the curve in two dimensions defined parametrically by for .

SECTION B

This section carries 60 marks and each question carries 20 marks. You may attempt all 4 questions but only marks for the best 3 questions will be counted. Begin each answer in this section on a fresh page. Write the number of the question at the top of each page.

Except in Question 1, all Maple functions and procedures must check that their arguments are of the correct types, and they should declare and use local variables where appropriate.

Write the Maple statements necessary to perform the following computations:

1. A point in two dimensions is defined, in terms of a parameter , to have Cartesian coordinates ( ) such that

, where .

(a) [3 marks] Load the plots package, and set the default for two-dimensional plotting so that the scaling is the same in all directions and no axes are plotted.

(b) [2 marks] Write a mapping r that implements the above definition of r as a function of .

(c) [2 marks] Write a mapping S that returns a plot of the locus of as defined above for .

(d) [2 marks] Write a mapping L that returns a plot of the straight line from the origin to the point .

(e) [2 marks] Write a mapping B that combines the curve S and the line L.

(f) [3 marks] Animate the line L tracing out the curve S for using a total of 32 frames.

(g) [3 marks] Plot the space curve in three dimensions defined in terms of a parameter to have Cartesian coordinates ( ) such that

, where ,

for , using normal axes through the origin.

(h) [3 marks] Plot the surface in three dimensions defined in terms of parameters to have Cartesian coordinates ( ) such that

,

for , where , and .

2. (a) [5 marks] Write a procedure called E2D that takes as its argument E a relative error that must be numerical and returns the corresponding value for Digits , by using exact arithmetic to compute and return the smallest positive integer d such that .

(b) [15 marks] Write a procedure called Newton that takes three arguments ( f , x0 , RelErr ), where f must be a mapping (taking one argument, which you are not required to check) and x0 and RelErr must be numerical values. The procedure should find a numerical approximation to a zero of f near to x0 to an estimated maximum relative error RelErr by implementing the following algorithm.

1. Determine and set an appropriate value for the precision of the numerical computation, based on the value of RelErr.

2. Compute the derivative of the mapping f .

3. Initialise the variable to the floating point approximation to x0.

4. Compute where f ' denotes the derivative of f .

5. If the magnitude of the difference between and relative to the magnitude of is smaller than RelErr then return the value of . (You may assume that will never be zero.)

6. Otherwise, assign the value of to the variable and repeat the algorithm from step 4.

3. [20 marks] Write a procedure called Area that takes two arguments ( y , var ), where y must be a polynomial with numerical coefficients and var must have the form , where x is a variable name and a and b are both numerical. The procedure should calculate a numerical approximation (accurate to the current default precision) to the total geometrical area bounded by the x -axis, the graph of y (a polynomial in x ) and the lines and . The procedure must regard all areas as positive, regardless of whether the graph is above or below the x -axis, by dividing the area into sub-areas bounded by segments of the graph that do not cross the x -axis and adding the magnitudes of these sub-areas. (You may assume that the Maple equation solvers return numerical solutions in increasing numerical order.)

4. [6 marks] Write procedure N2D that takes two arguments ( B , n ), where B must be a positive integer and n must be a non-negative integer, and returns the digit sequence that corresponds to the positional representation of n in base (radix) B . (You may assume B < 10.)

[8 marks] Write procedure D2N that takes one required argument B , which must be a positive integer, followed by a sequence of an arbitrary number of non-negative integers that represent the digits of the positional representation of n in base B , and returns the corresponding integer.

The procedures should work like this:

> N2D(10, 123);

> D2N(10, 1, 2, 3);

[6 marks] Write procedure

ConvertBase(DigitList, OldBase, NewBase)

that calls procedures N2D and D2N as appropriate to convert a list of digits representing a number in base OldBase to the list of digits representing the same number in base NewBase . Do this by applying N2D to the value returned by D2N , and converting between sequences and lists as necessary. The procedure must check that its arguments have appropriate types. It should work as follows.

> ConvertBase([1,2,3], 10, 2);

> ConvertBase([1,1,1,1,0,1,1], 2, 10);