5th lecture

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Elementary Programming using Mathcad (INF-20303)

• 5th lecture

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No lecture coming Wednesday

• Because of software problems many people run
  behind in the practicals
• We skip the material from last lecture
  practical

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• Series
• Example Fibonacci series
• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
  89,
• Prepare an efficient function that produces this
  series (or outcome)
• Therefore you need to detect the relation
  between successive terms
• The first two terms are 1
• Each next term is the sum of the two previous
  terms
• So the recursive relation is
• (Preparation of the function is an assignment in
  one of the
• coming practical sessions)

no vector needed
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• Another example
• Two issues are important
• Design an efficient function
• How many terms are needed for an accurate result?

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• numerator so
• a successive numerator is previous
  numerator
• denominator which
  means
• So

Denominator of next term is n times denominator of
previous term.

term n of series
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term n of series
Algorithm Make Repeat as long as
needed
value of series
value of a term
term number
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repeat
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repeat
for does that job
Algorithm ok However, specifying number of terms
isnot very attractive
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When to stop adding terms?

• Series to compute power of e
• Every successive term is smaller than its
  predecessor
• As of a certain term, the terms will not
  contribute to the
• value of the series (they are simply too
  small), for example
• Suppose r 2.5 and t 1.235 10-10
• To be able to add powers of 10 should be equal,
  so
• r 2.5 and t 0.0000000001235
• Suppose computer can store up to 7 decimal
  places, then
• 0.0000000001235 0

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• We may stop computing terms when t becomes 0
  relative to r
• At that time
• So repeat as long as

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A slightly different algorithm (using while)
t is second term
Repeat as long as
Make sure that from the first time holds.
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Try out (test) r 1 t x n 1
r 1 x n 1 1 2 t x x/2 x2/2
r 1 x x2/2 n 2 1 3 t (x2/2) x/3
x3/3!
r 1 x x2/2 x3/3! n 3 1 4 t
(x3/3! ) x/4 x4/4!
test at least 3 terms!
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Results


Results are incorrect for
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Another alternative
Not efficient and therefore not a correct
function! (dont do this during exam).
For term 3 the computer computes Overflow
when n is large inaccurate inefficient
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Why does that not work? At a certain point, we
have as numerator and denominator

• Remarks
• n! becomes large for small values of n, see 40!
  and 41!
• power also becomes large
• n! may become larger than the largest possible
  number
• n! increases faster than the power (terms
  decrease)
• outcome will be unreliable.
• the function given before is the correct function

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More series sinus

• numerator is previous numerator
• denominator
• denominator term 2 is denominator
  term 1
• denominator term 3 is denominator
  term 2
• so in general (in terms of n), denominator is
  times denominator previous term, with
• every term has opposite sign

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Algorithm

• numerator previous numerator
• denominator times previous
  denominator
• multiply every term by -1
• gives the algorithm

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More series

• numerator previous numerator
• denominator successively 1, 3, 5, 7, ..
• different situation denominator not by means of
  multiplication
• ? we need an additional variable

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Translate algorithm to Mathcad
Make
Repeat
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An example with several nested for statements
Which numbers lt 600 have this property?

• Solution
• Assume the number is abc
• the number consists of digits a, b, c
• the value of the number then is
• this value should be equal to
• which numbers of 3 digits lt 600 fulfill this
  condition?
• we check all possible numbers
• a varies from 1 up to and including 5
• b varies from 0 up to and including 9
• c varies from 0 up to and including 9

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Important element of algorithm

• We have to test the specific criterion

We have to check this for every possible
combination of a, b and c. Therefore we need
three nested for statements
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Structure of algorithm
if a number fulfills the requirement, we insert
the number in a vector.
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Algorithm
Because we store the numbers in a vector
We can do without h
sumThird has no argument, so it is not a function!
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Result is
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Preparation section 14.3, assignment 4
mod(m,n) returns remainder on dividing m by n We
may determine this by repeated subtractionfor
example mod(14,3) 14
3 11
3
8 3
5 3
2
Repeat subtraction of n until .? Apply a
statement to repeat the subtraction.
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Answer section 13.1, assignment 2
(write function to calculate area of polygon)
x1,y1
x2,y2
x0,y0
x4,y4
x3,y3
x5,y5
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Algorithm
What here?
last(x) can be an index
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Also possible factoring out factor 1/2
Or
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Can we expect any problems?
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Here we assume that the vectors x and y in the
call have the same number of elements. What if
we assume wrong?
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Function area can only return a correct result if
the number of elements of x and y are equal.
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Other solution The area equals to
x1,y1
x2,y2
x0,y0
x4,y4
x6,y6
x3,y3
x5,y5
Complexity of function reduces by introducing an
additional vertex
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(Extra vertex (point) is added to vectors
holding x and y values)
The area then can be computed with
Function then becomes
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(No Transcript)
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Answer section 13.1, assignment 3
Find the largest element in a matrix
Test the general case (max is largest element, x
is matrix)
We should test all elements of matrix x
rows(x) is number of rows of x. Because the
first row has number 0 rows(x)-1
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Recall maximum value in a vector. We started
there with 1. Is that correct here?