Try our Free Online Math Solver!

Math Assignment 2
One major approach to the problem of simplification of
mathematical expressions is to define
and maintain a canonical form valid over some interesting class of expressions
as elements
in that class are manipulated. In this problem set you will devise and analyze a
canonical form
for scseries (sinecosine series), a class of expressions we have invented for
these exercises, but
resembling socalled “Poisson” series used in Macsyma, CAMAL, and some other
systems .
Such series are typically highly useful in celestial mechanics (for orbital
calculations ) but are
sometimes used for (mathematically similar) differential equations problems that
occur in
other disciplines.
We’ll define scseries through the following sequence of
informal definitions (If you need
clarification or more formalism, ask):
• a variable is any alphanumeric name except for the
literal “
” which we reserved for
later use. Examples: x, y, t45.
• a rational number is an integer or a pair of integers, numerator and
denominator, in
some suitable notation.
• a term is a rational number times a variable. Note that unless it causes
confusion, we
will denote multiplication by juxtaposition of symbols . Examples:
,
3 t45.
• an arg is a sum of one or more terms . Example:
• a trigform is a rational number times a non negative power of the symbol times
sin
or cos of an arg. One example:
Another example:
• an scseries must be able to represent any sum of trigforms.
We will not formally specify the meaning of these
expressions, but they are meant to
correspond to the usual appliedmathematics notions of computations over
rational complex
numbers, indeterminates (to the extent you need to be aware of this, the
indeterminates
denote complex numbers , and not, say, matrices), and the trigonometric
functions. The
common interpretation of trig functions should tell you what we mean by
differentiation, for
example.
1. In order to make sure that any two equivalent
expressions are identical if expressed as
scseries, a number of other constraints must be placed upon the form. For
example we could
insist that the terms in the arg are in alphabetic order by variable name, and
there are no
duplicate names. What other constraints do you recommend, and why? (You may
interpret
this question as requiring you to eliminate the informality in the above
specifications.)
2. How do you propose to represent rational constants ?
How about zero? The constant
is not available in this representation. What would happen if it were?
3. In order to make some use of scseries, we need to
develop programs for the following
operations :
Given an scseries s, we must be able to compute the
derivative or the integral of s with
respect to some variable v. (The result must also be in canonical form, of
course.)
Given two scseries s and t, we must be able to compute
their sum, their product , and
their difference.
While you need not actually write these programs (although
you may if you wish), provide
enough of a pseudocode description to demonstrate that you’ve addressed all
important
mathematical or representation issues. In particular, you should characterize
the costs of
running these programs, stating any assumptions you are making along the way.
4. There is a transformation which preserves equivalence
only approximately turns out to
be of some use. Given a series s
, expand it in a new series s' where a particular variable,
say x is known to be a small quantity, . Then an integer degree d is chosen and
sin x and
cos x are expanded in power series to degree d. For example, if d = 3, then
sin(x + y) is
(approximately)
The dependence on x is changed from
trigonometric to algebraic. Explain how to do this. (Pseudocode algorithm,
cost)
5. Computing u := s^{4} can be done by computing t := s * s
and then u := t * t. Or it can
be computed by t := s * s, v := s * t, and finally u := s * v. Discuss which is
better. This
requires that you refer to your cost model of multiplication of scseries (in
answer to question
3) carefully. You will probably have to specify how you are storing scseries.
You should also
make some statements about the maximum, minimum, and/or likely sizes of s, t, u,
and v in
order to complete your analysis.
Prev  Next 