The GATE Grind

GATE 2022 BT – Question 30

Engineering Mathematics · Numerical Methods: Linear and nonlinear algebraic equations, trapezoidal and Simpson's rule, single step methods for ODEs · 1 mark · Numerical answer

For $x\ln x+x-1=0$, use Newton–Raphson with $x_0=3$. The value after one iteration (one decimal place) is ___________.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 1.29 to 1.31

Explanation

**Newton-Raphson method:**
$$x_{n+1}=x_n-\frac{f(x_n)}{f^\prime(x_n)}.$$

For $f(x)=x\ln x+x-1$: $f^\prime(x)=\ln x+1+1=\ln x+2$.

**At $x_0=3$:**
$$f(3)=3\ln3+3-1=3(1.0986)+2=5.2958,\qquad f^\prime(3)=\ln3+2=3.0986 .$$

$$x_1=3-\frac{5.2958}{3.0986}=3-1.7091=1.2909\approx\mathbf{1.3}.$$