Numerical Analysis Questions and Answers 2025/2026
Taylor's Theorem with Remainder - CORRECT ANSWER-
Fixed Point Convergence - CORRECT ANSWER-If g is continuously differentiable,
g(r)=r, and ‖g'(r)‖<1, then the fixed point iteration converges linearly with rate S to
the fixed point r.
S=‖g'(r)‖
Newton's Method - CORRECT ANSWER-Approximates the root of non-linear
functions by finding the root of the tangent line.
In the second iteration, plug in the approximation from the first iteration as "x_n".
Repeat recursively.
Newton's Method Convergence - CORRECT ANSWER-Newton's Method converges
quadratically to a root x if f'(r)≠0 at rate M.
M=lim(e_(i+1))/e_i²
Derivation of Newton's Method - CORRECT ANSWER-f'(x_0)(x-x_0)=0-f(x_0)
x-x_0=-f(x_0)/f'(x_0)
, x=x_0-f(x_0)/f'(x_0)
Secant Method - CORRECT ANSWER-Approximates the root of the function by
taking the secant line through the last two approximation. The new guess is the
point in which the secant line intersects the x-axis.
Start with two initial approximations p₀ and p₁. The next guess, p₂, is the x-
intercept of the line joining (p₀,f(p₀) and (p₁,f(p₁).
The approximation p₃ is the x-intercept of the line joining (p₁,f(p₁)) and (p₂,f(p₂)),
and so on.
Method of False Position - CORRECT ANSWER-Combination of the Secant Method
and the Bisection Method. It uses the Bisection Method to ensure that the two
previous approximations used for the secant line have opposite signs.
Choose initial guesses p₀ and p₁ with opposite sides (p₀*p₁ < 0).
The next guess, p₂, is the x-intercept of the line joining (p₀,f(p₀) and (p₁,f(p₁).
Use p₀ and p₂ or p₁ and p₂ as your next guesses. Use p₀ and p₂ if f(p₀) and f(p₂)
have opposite signs. Else, use the other option.
Continue until f(p_n)=0.
Taylor's Theorem with Remainder - CORRECT ANSWER-
Fixed Point Convergence - CORRECT ANSWER-If g is continuously differentiable,
g(r)=r, and ‖g'(r)‖<1, then the fixed point iteration converges linearly with rate S to
the fixed point r.
S=‖g'(r)‖
Newton's Method - CORRECT ANSWER-Approximates the root of non-linear
functions by finding the root of the tangent line.
In the second iteration, plug in the approximation from the first iteration as "x_n".
Repeat recursively.
Newton's Method Convergence - CORRECT ANSWER-Newton's Method converges
quadratically to a root x if f'(r)≠0 at rate M.
M=lim(e_(i+1))/e_i²
Derivation of Newton's Method - CORRECT ANSWER-f'(x_0)(x-x_0)=0-f(x_0)
x-x_0=-f(x_0)/f'(x_0)
, x=x_0-f(x_0)/f'(x_0)
Secant Method - CORRECT ANSWER-Approximates the root of the function by
taking the secant line through the last two approximation. The new guess is the
point in which the secant line intersects the x-axis.
Start with two initial approximations p₀ and p₁. The next guess, p₂, is the x-
intercept of the line joining (p₀,f(p₀) and (p₁,f(p₁).
The approximation p₃ is the x-intercept of the line joining (p₁,f(p₁)) and (p₂,f(p₂)),
and so on.
Method of False Position - CORRECT ANSWER-Combination of the Secant Method
and the Bisection Method. It uses the Bisection Method to ensure that the two
previous approximations used for the secant line have opposite signs.
Choose initial guesses p₀ and p₁ with opposite sides (p₀*p₁ < 0).
The next guess, p₂, is the x-intercept of the line joining (p₀,f(p₀) and (p₁,f(p₁).
Use p₀ and p₂ or p₁ and p₂ as your next guesses. Use p₀ and p₂ if f(p₀) and f(p₂)
have opposite signs. Else, use the other option.
Continue until f(p_n)=0.
