7th lecture

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Elementary Programming using Mathcad (INF-20303)

• 7th lecture

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• Finding roots Rule of False position (Regula
  Falsi)
• Consider interval
• Given and
• so 1 or more roots on
  interval

b
a
c

• Note its not always the same side of the
  interval
• In some cases it will select only the left part,
  sometimes only the right part of the interval,
  sometimes alternating (see next slide).

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In interval point of inflection
b
1
a
2
Consider this when preparing the function
RegulaFalsi!
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Algorithm
Rule of False position converges always,
sometimes slow.
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• Finding roots Newton Raphson
• Select a point on the graph of f(x)
• Draw the tangent in that point
• Determine intersection of the tangent with
  x-axis
• Repeat procedure in the intersection until root
  is accurate

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p1
p2
p0
Recursive formula
Problem?
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Newton Raphson does not converge always, for
example
Start x0 gives
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Fixed point iteration
Suppose you want a root of
So
Or
and
(equivalent form)
Fixed point iteration works as follows
x0 4 x1 3.31662 x2 3.10375 x3 3.01144 .
. . xn 3.0
Start value is
Compute successively
etc. Hopefully it will converge.
Stop when
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In general, other equivalent forms can be
derived, e.g.
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• Iterations
• Sometimes iterations do not converge
• An iteration is of form
• (F does not need to be a mathematical
  function!)
• Convergence requires for all
• This should also hold for
• If iteration does not converge try another
• starting point.

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• Advantages of iterations seen from Computer
  Science perspective
• Iterations are simple to program
• If iteration converges then arithmetic errors
  will be resolved
• in the next iteration
• Suppose you need to know the roots of
• Which method is better?
• quadratic-formula
• iterative method like Rule of False position

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Stop criteria
E.g. in the Newton iteration (sqroot) to
compute , we used criterion tol.

• Suppose stop criterion is
• absolute
• relative
• mixed

e0.01
When a 1 then tolerance 1. When a 10 then
tolerance 0.1 For high values rather stringent
Stringent for small values, better suited for
high values
For small values of a ad is small absolute
criterion For high values of a e is negligible
relative criterion
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Answer practical 6, section 15.1 assignment 2

• Function will not work when b 0! (we have to
  adapt function again)
• If explicit or implicit prerequisites about
  arguments may be not
• fulfilled

TEST and generate ERROR message!
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Answer practical 6, section 15.1, assignment 3
prepare a function that finds all divisors of a
given number, the number itself excluded.
What is the test in this function? Assume N is
the number. We can test whether N can be divided
by 2 by
(mod is applied here instead of function
remainder)
More generally is i a divisor of N?
To find all divisors on N, repeat this
(Why not a while?)
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When a divisor is found, it should be stored in
a vector, e.g. v (here)
No need to test N itself
j indicates position in vector
Make room for the next number
vóór de for-lus
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Do we need all N-1?
is sufficient
Result vector v NOT
Assignment indicated that Ngt0.
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(Assignment was Give divisors lt N, so this is
actuallynot correct)
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Suppose we take advantage of the knowledge that 1
is always a divisor and insert 1 into the vector
without further testing
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(No Transcript)
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mod(1,2) ? 0
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N 1 not correct We need yet another
error message!
Conclusion sometimes an optimization leads to a
complicated function.
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• Divide and conquer
• Suppose one needs to prepare a function to find
  the first 100
• prime numbers
• Prepare two functions
• one function that tests whether a number is
  prime
• another function that finds the first 100 prime
  numbers
• In this second function you call the first
  function.
• Separate responsibilities, resulting a function
  that is clear and can be maintained more easily

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Answer section 15.1, assignment 4 Series to
compute sine
Make
Repeat as long as needed
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Test your function
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Correct
Not correct
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Do not
The magnitude of t related to the value of r is
of importance!