Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 29.4, Problem 3E
Program Plan Intro
To calculatethe dual of maximum-cost-flow linear program as given in lines (29.47) − (29.50) on page 860 and to explain how to interpret the given problem
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We have learned the mid-point and trapezoidal rule for numercial intergration in the tutorials. Now you are asked to implement the Simpson rule, where we
approximate the integration of a non-linear curve using piecewise quadratic functions.
Assume f(x) is continuous over [a, b] . Let [a, b] be divided into N subintervals, each of length Ax, with endpoints at P = x0, x1, X2, ..., Xn,..., XN.
Each interval is Ax =
(b – a)/N.
The Simpon numerical integration rule is derived as:
N-2
Li f(x)dx =
* f(x0) + 4 (2n odd f(xn)) + 2 ( En=2,n even
N-1
f(x,) + f(xn)] .
Now complete the Python function InterageSimpson(N, a, b) below to implement this Simpson rule using the above equation.
The function to be intergrate is f (x) = 2x³ (Already defined, don't change it).
In [ ]: # Complete the function given the variables N,a,b and return the value as "TotalArea".
# Don't change the predefined content, only fill your code in the region "YOUR CODE"
from math import *
def InterageSimpson (N, a, b): # n is…
We have learned the mid-point and trapezoidal rule for
numercial intergration in the tutorials. Now you are asked to
implement the Simpson rule, where we approximate the
integration of a non-linear curve using piecewise quadratic
functions.
Assume f(x) is continuous over [a, b] . Let [a, b] be
divided into N subintervals, each of length Ax, with
endpoints at P = x0, x1, x2,..
Xn,..., XN. Each
interval is Ax = (b − a)/N.
The equation for the Simpson numerical integration rule is
derived as:
f f(x) dx
N-1
Ax [ƒ(x0) + 4 (Σ1,n odd f(xn))
ƒ(x₂)) + f(xx)].
N-2
+ 2 (n=2,n even
Now complete the Python function InterageSimpson (N, a,
b) below to implement this Simpson rule using the above
equation.
The function to be intergrate is ƒ(x) = 2x³ (Already
defined in the function, no need to change).
USING PYTHON
A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1).
Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added.
Please show steps and explain.
Chapter 29 Solutions
Introduction to Algorithms
Ch. 29.1 - Prob. 1ECh. 29.1 - Prob. 2ECh. 29.1 - Prob. 3ECh. 29.1 - Prob. 4ECh. 29.1 - Prob. 5ECh. 29.1 - Prob. 6ECh. 29.1 - Prob. 7ECh. 29.1 - Prob. 8ECh. 29.1 - Prob. 9ECh. 29.2 - Prob. 1E
Ch. 29.2 - Prob. 2ECh. 29.2 - Prob. 3ECh. 29.2 - Prob. 4ECh. 29.2 - Prob. 5ECh. 29.2 - Prob. 6ECh. 29.2 - Prob. 7ECh. 29.3 - Prob. 1ECh. 29.3 - Prob. 2ECh. 29.3 - Prob. 3ECh. 29.3 - Prob. 4ECh. 29.3 - Prob. 5ECh. 29.3 - Prob. 6ECh. 29.3 - Prob. 7ECh. 29.3 - Prob. 8ECh. 29.4 - Prob. 1ECh. 29.4 - Prob. 2ECh. 29.4 - Prob. 3ECh. 29.4 - Prob. 4ECh. 29.4 - Prob. 5ECh. 29.4 - Prob. 6ECh. 29.5 - Prob. 1ECh. 29.5 - Prob. 2ECh. 29.5 - Prob. 3ECh. 29.5 - Prob. 4ECh. 29.5 - Prob. 5ECh. 29.5 - Prob. 6ECh. 29.5 - Prob. 7ECh. 29.5 - Prob. 8ECh. 29.5 - Prob. 9ECh. 29 - Prob. 1PCh. 29 - Prob. 2PCh. 29 - Prob. 3PCh. 29 - Prob. 4PCh. 29 - Prob. 5P
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