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This course will introduce the mathematical foundations of numerical analysis and illustrate its importance for various problems in sciences. Topics will include but may not be limited to: floating-point representation of numbers and computer arithmetic; root-finding algorithms and rate of convergence; the bisection method; fixed-point iterations; Newton's method and the secant method; error analysis of iterative methods; polynomial interpolation; numerical differentiation; numerical integration; initial value problems (IVPs) for ordinary differential equations (ODEs); Euler's method; higher-order Taylor methods; local truncation errors; Runge-Kutta methods; stability analysis; systems of differential equations; basics of gradient descent; Newton's method in N dimensions; boundary value problems (BVPs) for ordinary differential equations (ODEs); linear and nonlinear shooting; finite-difference methods for linear and nonlinear problems; time permitting, an introduction to finite differences solutions to partial differential equations (PDEs).