Which of the following statements are true and which are false? Give a

Question

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:

$y_{n+1}=\left(1+\lambda h+\frac{\lambda^2 h^2}{2}\right)y_n,\quad\lambda>0,\quad n\geq 0$

is relatively stable.

v) The method:

$x_{n+1}=\frac{x_n}{2}+\frac{9}{8x_n}$

converges to 1.5 for any choice of initial approximation.

08 Feb 2025

Answer :

Word Count : 242

i) True. Let $f(x)=x^3-4x-16$. Then $f(3)=27-12-16=-1$ and $f(4)=64-16-16=32$. Since $f$ is continuous and $f(3)$ and $f(4)$ have opposite signs, by the Intermediate Value Theorem there is at least one root of $f(x)=0$ in $[3,4]$. ii) False. The secant method has order $p$ satisfying the equation $p^2=p+1$. Solving gives $p=\frac{1+\sqrt5}{2}\approx1.618$. Therefore the order is about $1.618$, not $0.62$. iii) False. The ________ ______ ____ _______ _______ ______ _____ ______ ____.
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