Estimate the eigen values of the matrix:
using the Gerschgorin bounds. Also, draw the rough sketch of the region in which the eigen values lie.
To estimate the eigenvalues of a matrix using Gerschgorin bounds, we consider each row of the matrix and construct discs in the complex plane. If the matrix is $$ A = [a_{ij}], \quad i,j = 1,2,\dots,n $$ then for each row $i$, the Gerschgorin disc is centered at $a_{ii}$ with radius $$ R_i = \sum_{j \neq _________ ____ ________ _____ __________ __________ __________ _____ _________ ______ __________ _____.
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Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:
i) The equation x3 4x 16 = 0 has a root in the interval [3, 4].
ii) The order of convergence of the secant method is 0.62.
iii) For the system of linear equations:
5x + y + 2z = 34
4y − 3z = 12
10x − 2y + z = −4
the matrix is diagonally dominant.
iv) The numerical method:
is relatively stable.
v) The method:
converges to 1.5 for any choice of initial approximation.
Define a subgroup of a group. Check whether
is a subgroup of the the group of 2 × 3 matrices over ℂ under addition
Check whether or not (3) is a maximal ideal in Z9.
Let . Check whether Check whether R is a subring of M₂(R). Is R an ideal of M₂(R)? Justify your answer.
Calculate the nth divided difference of 1/x, on the nodal points X0 , X1 , ......., Xn .
Given the data:
f(3) = 168,
f(7) = 120
and f(9) = 72
If f(k) is estimated as 138 using the Newton’s form of the interpolating polynomial, then find the value of .k
Find one root of the equation:
x3 - 2x 5 = 0
in the interval [3,2] using Birge-Vieta method. Perform only one iteration.