Engineering Mathematics and Numerical Methods
Engineering Mathematics and Numerical Methods tests linear algebra, calculus, differential equations, probability, and numerical algorithms. Expect setup, convergence, and error bounds more than long manual computation.
Core rules
- A square system Ax=b is solvable uniquely when det(A)≠0; ill-conditioning amplifies input error.
- Newton-Raphson uses x_{n+1}=x_n-f(x_n)/f'(x_n); it converges quadratically near a simple root.
- Trapezoidal rule error scales as h²; Simpson's rule needs an even number of intervals and scales as h⁴.
- For eigenvalues, check trace and determinant as quick invariants.
- Numerical differentiation amplifies noise; integration smooths it.
Worked example
Estimate √2 by Newton-Raphson with f(x)=x²-2, f'(x)=2x, starting x0=1.5. x1=1.5-(2.25-2)/3=1.5-0.08333=1.41667. x2=1.41667-(2.00694-2)/2.83333=1.41667-0.00245=1.41422. The true value is about 1.41421, so two iterations give four correct decimals. The rapid drop in error illustrates quadratic convergence when the derivative is nonzero and the start is close enough.
Traps
Using degrees in trigonometric derivatives; applying Simpson's rule with an odd interval count; stopping Newton iterations without an error criterion; ignoring pivot selection in Gaussian elimination; and treating a high condition number as a small residual problem.