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Quantitative Methods
Using the Horner’s method find the values of f(4) and f'(4) for the polynomial
f(x) = x^4 +2x^3 −x^2 +1.
f(x) = x^4 +2x^3 −x^2 +1.
Quantitative Methods
From the data
x 1 1.5 2 2.5 3
f(x) 1 .5 1.3 .8 1.5
interpolate the value of f(2.8) using the Newton’s backward difference formula.
x 1 1.5 2 2.5 3
f(x) 1 .5 1.3 .8 1.5
interpolate the value of f(2.8) using the Newton’s backward difference formula.
Quantitative Methods
Find a root of the equation 3x^3+10x^2+10x+7 = 0 which is close to −2.0, using
the Berge -Vieta method. Perform two iterations of the method.
the Berge -Vieta method. Perform two iterations of the method.
Quantitative Methods
Consider the following data
x 1.0 1.3 1.6 1.9 2.2
f(x) 0.76519 0.62008 0.45540 0.28181 0.11036
Use stirling’s formula to approximate f(1.5) with x0 = 1.6
x 1.0 1.3 1.6 1.9 2.2
f(x) 0.76519 0.62008 0.45540 0.28181 0.11036
Use stirling’s formula to approximate f(1.5) with x0 = 1.6
Quantitative Methods
Compute integral 0 to 4, f(x)dx using the Romberg integral technique on the trapezoidal integrals evaluated by the trapezoidal rule taking h=1 and h=0.5.The tabulated values are given below.
x 0 0.5 1 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 1 4 3 2 2.5 2.9 3.6 4 1.8
x 0 0.5 1 1.5 2.0 2.5 3.0 3.5 4.0
f(x) 1 4 3 2 2.5 2.9 3.6 4 1.8
Quantitative Methods
From the values of f(x) = xex given in the table
x 1.8 1.9 2.0 2.1 2.2
f(x) 10.8894 12.7032 14.7787 17.1489 19.8550
find f
00(2.0) using the central difference formula of O(h
2
) for h = 0.1 and h = 0.2.
Calculate T.E. and actual error.
x 1.8 1.9 2.0 2.1 2.2
f(x) 10.8894 12.7032 14.7787 17.1489 19.8550
find f
00(2.0) using the central difference formula of O(h
2
) for h = 0.1 and h = 0.2.
Calculate T.E. and actual error.
Quantitative Methods
Taking the endpoints of the last interval obtained in f(x) = x^3−5x^2 +1 = 0 as the initial
approximations, perform two iterations of the secant method to approximate the
root.
approximations, perform two iterations of the secant method to approximate the
root.
Quantitative Methods
Find the interval of unit length that contains the smallest positive root of the
equation f(x) = x
3 −5x
2 +1 = 0. Starting with this interval, find an interval of
length 0.05 or less that contains the root, by Bisection method.
equation f(x) = x
3 −5x
2 +1 = 0. Starting with this interval, find an interval of
length 0.05 or less that contains the root, by Bisection method.
Quantitative Methods
Find the inverse of the matrix
1 −1 4
2 9 8
6 5 2
using LU decomposition method.
1 −1 4
2 9 8
6 5 2
using LU decomposition method.
Quantitative Methods
Perform four iterations of the inverse power method to compute the smallest
eigenvalue in magnitude, and the corresponding eigenvector of the matrix A given below above.
A=1 −1 1
2 0 3
1 4 −1
eigenvalue in magnitude, and the corresponding eigenvector of the matrix A given below above.
A=1 −1 1
2 0 3
1 4 −1
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#339914 on Dec 2023
#339914 on Dec 2023