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Programming with Python in
Numerical Analysis
by Matthew L. Greiss
The ﬁeld of numerical analysis makes use of many diﬀerent algorithms in order to
approximate the solutions for a wide variety of problems. A number of these algorithms
can be quite long and may require several calculations. Previously, calculations such as
these had to be done by hand, and could thus take minutes, hours, or even days to
complete. Now, with the advent of modern programming, devices such as computers,
tablets, and even phones, can complete these calculations in a matter of seconds. The
programs that are used to perform such calculations can be written in a number of
diﬀerent languages, one of which is Python.
This paper will cover ten programs that have been written in the programming
language Python and are designed to solve problems or complete tasks within the ﬁeld of
numerical analysis. The main point of this paper will be to examine the methodology
behind each program and how the source code executes upon those ideas. Since this
paper’s main topic is the source code of these programs, a pdf of it will accompany this
paper. It would be unreasonable to cover every character of the source code, so
insigniﬁcant elements, as well as those whose purpose is immediately apparent, will not be
discussed. While the accompanying pdf will contain the entirety of the source code, it will
still be helpful to include excerpts from it, as this will make certain explanations clearer.
When source code is presented it may be abridged or slightly altered for the sake of clarity.
1
�The ﬁrst program will convert binary machine numbers into base10 numbers. These
are numbers that computers use to store base10 numbers. The second, third, fourth, and
ﬁfth programs, are all rootﬁnding algorithms. The second is a bisection method program,
and it repeatedly bisects intervals until they converge upon a root. The third is a
ﬁxedpoint iteration program and it will isolate one of a function’s variables, and ﬁnd a
ﬁxed point for the resulting expression. The fourth is a NewtonRaphson method program
and it will iterate an expression that is comprised of both a function and its derivative.
The ﬁfth is a secant method program and it will iterate a recurrence relation. The sixth is
an Euler method program, and it will solve ordinary diﬀerential equations by generating a
series of discrete points. The ﬁnal four programs all use interpolation to approximate
functions. The seventh is a Lagrange polynomial program and it produces a polynomial by
generating a series of terms, each constructed from a combination of data points, and
adding them together. The eighth is a Newton polynomial program and it produces a
polynomial by taking the divided diﬀerences of adjacent data points. The ninth is a
Hermite interpolation program and it approximates coordinates of a function by taking the
divided diﬀerences of a function’s data points and its derivatives data points. The tenth
and ﬁnal program is a cubic spline interpolation program, and it produces a piecewise
function, comprised of cubic splines, in order to approximate functions.
Speciﬁcally, they are the bisection method, ﬁxedpoint iteration, NewtonRaphson method,
and secant method.
2
�
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Greiss, Matthew L.
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5/1/2020
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Programming with Python in Numerical Analysis
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Mathematics
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Mathematics