Exercise: error of the sin x ≈ x approximation
Computing the numerical and percentage error when approximating by .
Solved exercises
Worked problems with concrete data and their detailed solution.
Foundations
Exercise: iterative error with e^0.5
Approximating by its Taylor series, adding terms until the percentage iterative error drops below 0.05 %.
Exercise: Taylor of cos(x) up to order 3
Taylor series expansion of cos(x) around zero, taking terms up to order 3.
Interpolation
Exercise: Hermite with the Bessel function
Full construction of the degree-5 Hermite polynomial with three nodes to approximate , step by step.
Exercise: Newton with population data
Full divided-difference table for the 1971–2011 census and estimate of the 2005 population with the degree-4 Newton polynomial.
Exercise: Lagrange with population data
Building the Lagrange basis functions for the 1971–2011 census and estimating the 2005 population, compared with Newton.
Differentiation
Exercise: comparing derivative formulas
Computing for with the six finite-difference formulas and comparing their errors against the exact value .
Exercise: Richardson on ln(x)
Approximating for with forward differences and improving to via Richardson extrapolation.
Integration
Exercise: trapezoid and Simpson on a smooth integral
Approximation of the integral of from 0 to with 4 and 8 subintervals, comparing errors.
Exercise: trapezoid and midpoint
Comparing composite trapezoid, simple midpoint and composite midpoint, with absolute and relative errors.
Exercise: work from tabulated data
Computing work by integrating F(x)cos(alpha(x)) from a table, using trapezoid, Simpson and midpoint.
Exercise: Gauss-Legendre on a general interval
Changing variables from [1,1.5] to [-1,1] and applying Gauss-Legendre with n=2 and n=3.
Exercise: Gauss-Chebyshev and an error bound
Determining how many nodes guarantee six decimals in an integral with Chebyshev weight.
Exercise: double integral with Gauss-Legendre
Transforming a rectangle to [-1,1]×[-1,1] and solving a double integral with n=m=3.
Exercise: double integral of a surface
Comparison between double Simpson and Gauss-Legendre for an integral derived from the hemisphere .
ODEs
Exercise: Euler and RK4 by hand on a first-order ODE
Solve for y'=f(t,y) in an ODE given in implicit form and approximate y(3) with two Euler steps and one RK4 step, comparing both results.
Exercise: second-order ODE as a system, by hand
Reduce y''−sin y=0 to a first-order system and approximate y(3) with two Euler steps and one vector RK4 step.
Exercise: numerical order estimation (Euler, Heun, RK4)
On the same logistic IVP, Euler's order is estimated with the exact solution, Heun's by comparing successive meshes without an exact solution, and RK4's, confirming orders 1, 2 and 4.
Exercise: stability of explicit and implicit Euler
Full analysis of : the amplification factor of each method, the explicit method's stability condition , the unconditional stability of the implicit one and a numerical check with .
Exercise: the SIR model with Euler, Heun and RK4
Integration of the SIR epidemic system with the three one-step methods on the same mesh, comparing how the method's order visibly changes the results.
Exercise: AB2 and order estimation
Solving a logistic IVP (Verhulst) with AB2 started with Heun and estimating the order numerically by doubling the number of subintervals.
Exercise: explicit versus predictor-corrector
Comparing the maximum error of AB2, AB4 and the predictor-correctors ABM2 and ABM4 on the same Verhulst IVP.
Linear systems
Exercise: Jacobi and Gauss-Seidel compared
Solving a 4×4 system with Jacobi and with Gauss-Seidel from x⁰=0, comparing how many iterations each needs for the same tolerance.
Exercise: convergence by spectral radius
A non-diagonally-dominant matrix where Jacobi diverges but Gauss-Seidel converges, decided by computing the spectral radius of each iteration matrix.
Nonlinear equations
Exercise: bisection by hand
Six bisection iterations for cos²x−x=0 on [0,1], with the chain of intervals, the error bound at each step and the prediction of the required number of iterations.
Exercise: Newton on x=cos²x
Full application of Newton's method to x=cos²x from x0=0.3: table of iterates, residuals, increments and ACOC tending to the theoretical order 2.
Exercise: the secant by hand
The secant method applied to cos²x−x=0 from x0=0, x1=1: five iterations with explicit divided differences and superlinear convergence on display.
Exercise: numerical comparison of iterative methods
Newton, Halley, Ostrowski, Traub, midpoint, Jarratt and double Newton on test functions with tolerance 10⁻¹⁰⁰: iterations, residuals and ACOC confirming the theoretical orders.
Nonlinear systems
Exercise: Newton for a system, by hand
Two Newton steps by hand on the system , : building the Jacobian, solving each step's linear system and visible quadratic convergence towards .
Exercise: Newton on a 2×2 system with iteration table
Full solution of , with Newton from : table of iterates, residual and increment norms, and ACOC settling at 2.
Exercise: numerical comparison on systems
Newton, Trapezoids, Golden Ratio, NA, Jarratt and RN on two test systems with tolerance 10⁻¹²: iterations, norms and ACOC, with RN (order 6) as the most effective method.