State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.
a) The equation has not root in the interval [3, 5].
b) , where E is the shift operator and
is the forward difference operator.
c) Every system of linear equations can be solved using the LU decomposition method.
d) For the data (2, 4), (1, 5), (3, 6) the Newton's divided difference f[x0, x1, x2] is .
e) The Newton-Raphson method cannot be used to find a cube root of a positive real number.
a) True – If the function (f(x)) is continuous on [3, 5] and the given equation does not change sign in this interval, by Bolzano’s theorem (Intermediate Value Theorem), it has no root there. Counter-example: ___ _______ _________ ________ __________ ______ ______ ______ ____.
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Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.
) Evaluate by using trapezoidal rule with
and
. Use Romber's method to find the best value of
.
Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations
Show that the method is convergent and hence find its rate of convergence.
) Evaluate by using trapezoidal rule with
and
. Use Romber's method to find the best value of
.
) Estimate the eigenvalues of the matrix
using the Gerschgorin bounds.
विश्रामावस्था से आरंभ कर रही एक गाड़ी का वेग पहले घंटे के लिए निम्नलिखित तालिका में दिया गया है। सिम्पसन
का नियम लागू करके, इस घंटे में गाड़ी द्वारा तय की गई दूरी ज्ञात कीजिए :
Find the missing values in the following table:
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 0 | 2 | - | 18 | - | 90 |
सिद्ध कीजिए कि ।