Ashmit Rao

Derivatives, partial derivatives, differentiation with respect to a vector, gradient, Hessian and Jacobian. Taylor's theorem in 1 dimension (Lagrange remainder), Taylor's theorem in $n$ dimensions (remainder only briefly). Examples.

Optimization in 1 and $n$ dimensions. Classification of turning points via Taylor's theorem. Convexity. Constrained optimization: Lagrange multipliers. Examples.

Numerical integration in 1 dimension: midpoint and Simpson's rules, complexity and error analysis. Briefly, integration in $n$ dimensions and Monte Carlo methods. Examples.

Floating-point numbers and rules of thumb for accuracy in practice. Convergence rates of iterative methods.

Numerical root finding in 1 dimension by bisection, Newton's method, secant method. Root finding in $n$ dimensions by Newton's method, and briefly quasi-Newton methods. Complexity and error analysis. Examples.

Numerical optimization in 1 and $n$ dimensions: root finding for gradient and gradient descent methods. Complexity and error analysis. Examples.

Applications.