Syllabus for weeks 1-7: basic concepts (errors, fixed points of maps); locating roots in 1-d (fixed-point iteration, bisection method, false position, secant method, Newton-Raphson method, Muller's method); solving systems of nonlinear equations (fixed-point iteration, Newton's method); solving systems of linear equations (Gaussian elimination and LU factorization, Jacobi method, Gauss-Seidel method, successive over-relaxation); finding eigenvalues of matrices (power method, QR factorization); function approximation (least squares, continuous least squares, Fourier method, interpolation [polynomial, Hermite, spline])

Recommended textbook: my lecture notes! Otherwise, any numerical analysis textbook will cover most things, but will also contain many details that are not needed for this course.