Numerical Methods: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

Numerical methods

Bisection: c = (a + b)/2, width after n steps (b − a)/2ⁿ
Newton–Raphson: x_n+1 = xₙ − f(xₙ)/f′(xₙ)
Linear interpolation: y = y₁ + (y₂ − y₁)(x − x₁)/(x₂ − x₁)
Trapezoid rule: T = (h/2)[f₀ + 2f₁ + … + 2f_n−1 + fₙ], h = (b − a)/n
Simpson’s rule (n even): S = (h/3)[f₀ + 4f₁ + 2f₂ + … + 4f_n−1 + fₙ]
Trapezoid error bound: |E| ≤ (b − a)h²M/12, M = max|f″|
Euler: y_n+1 = yₙ + h·f(tₙ, yₙ)
Heun: y_n+1 = yₙ + (h/2)(k₁ + k₂), k₂ = f(tₙ + h, yₙ + hk₁)
RK4: y_n+1 = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)

Method

For a root: check the sign change for bisection, or compute f and f′ for Newton, and keep six or more digits in a table of iterates.
For a table value: interpolate linearly between the two neighbouring points.
For an integral: find h = (b − a)/n, list f at every node, apply the weights (1, 2, …, 2, 1 for trapezoid; 1, 4, 2, …, 4, 1 for Simpson with even n) and multiply by h/2 or h/3.
For an ODE: step from t₀ with yₙ₊₁ = yₙ + h·f(tₙ, yₙ) (Euler) or the averaged slope (Heun), one row per step, until you reach the requested t.

Sequences & series

Arithmetic sequence: aₙ = a₁ + (n − 1)d, Sₙ = n(a₁ + aₙ)/2
Geometric sequence: aₙ = a₁rⁿ⁻¹, Sₙ = a₁(1 − rⁿ)/(1 − r)
Infinite geometric series: S = a₁/(1 − r), |r| < 1
p-series: Σ1/kᵖ converges only for p > 1
Ratio test: L = lim |a_k+1/a_k|: L < 1 converges, L > 1 diverges
Radius of convergence: R = lim |c_k/c_k+1|
Taylor polynomial: Pₙ(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ/k!
Maclaurin series: eˣ = Σxᵏ/k!, sin x = x − x³/3! + …, cos x = 1 − x²/2! + …, ln(1 + x) = x − x²/2 + …
Lagrange error bound: |Rₙ| ≤ M·|x − a|ⁿ⁺¹/(n + 1)!

Method

Identify the type: arithmetic (constant difference), geometric (constant ratio), a standard series, a power series or a Taylor polynomial.
For sums use Sₙ = n(a₁ + aₙ)/2, Sₙ = a₁(1 − rⁿ)/(1 − r), or S = a₁/(1 − r) when |r| < 1, with a₁ the first term of the sum.
For convergence: n-th term test first, then compare with Σ1/kᵖ or a geometric series, and use the ratio or root test for factorials and k-th powers.
For Taylor polynomials list f(a), f′(a), f″(a), …, divide the k-th derivative by k!, and bound the error with M·|x − a|ⁿ⁺¹/(n + 1)!.

Matrices & determinants

2×2 determinant: det = ad − bc
2×2 inverse: A⁻¹ = (1/det)·[d −b; −c a]
Product: (AB)_ij = Σ_k A_ikB_kj
Eigenvalues: det(A − λI) = 0
Cramer’s rule: x_i = det(A_i)/det(A)

Differential equations

Exponential growth/decay: dy/dt = ky → y = y₀e^kt
Half-life: k = ln 2 / t_½
Newton's law of cooling: T(t) = T_a + (T₀ − T_a)e^−kt
Integrating factor: y′ + p(x)y = q(x): μ = e^{∫p dx}, (μy)′ = μq
Mixing tank: dm/dt = r·c − (r/V)·m

2Practice problems

1.Numerical methodsMedium

Approximate ∫₀³ √x·e^−x dx with Simpson's rule using n = 2 strips. Give S.

f(x) = √x·e^−x[a, b] = [0, 3]n = 2

S =

2.Numerical methodsHard

Solve y′ = y − t², y(0) = 2 with Euler's method and step h = 0.2 up to t = 0.8. The exact solution is y = t² + 2t + 2 + (y₀ − 2)eᵗ. Find the global error |E| = |y(0.8) − y₄|.

y′ = y − t²y(0) = 2h = 0.2

|E| =

3.Sequences & seriesMedium

Find the sum S = Σ_k=1^∞ 4/(k(k + 1)).

a_k = 4/(k(k + 1))

S =

4.Sequences & seriesHard

f(x) = eˣ is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 8, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣx = 2M = 8ε = 1 × 10⁻⁵

n =

5.Matrices & determinantsMedium

Find the determinant det A of the 3×3 matrix A shown.

A = [−2 5 2; 5 −2 −3; 3 −1 3]

det A =

6.Matrices & determinantsHard

Find the larger eigenvalue λ of the matrix A shown.

A = [5 4; −3 −3]

λ =

7.Differential equationsMedium

Solve y″ + 5y′ + 6y = 0 with y(0) = 5 and y′(0) = 6. Find y(2).

y″ + 5y′ + 6y = 0y(0) = 5y′(0) = 6

y(2) =

8.Differential equationsHard

Solve the initial value problem y′ + 2y/x = 3 for x > 0, with y(1) = 4. Find y(3).

y′ + 2y/x = 3y(1) = 4

y(3) =

3Solutions

1.Numerical methods

Answer: S = 0.5897

Numerical integration: split [a, b] into n strips of width h = (b − a)/n with nodes x₀ … xₙ. Trapezoid rule: T = (h/2)[f₀ + 2f₁ + 2f₂ + … + 2f_n−1 + fₙ]. Simpson's rule (n even): S = (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + … + 4f_n−1 + fₙ]. The trapezoid error shrinks like h² (bound (b − a)h²·M/12 with M = max|f″|), Simpson's like h⁴, so Simpson is usually far more accurate for the same nodes.

Nodes and values

h = 1.5

Σ wᵢfᵢ = 0 + 4·0.273278 + 0.0862337 = 1.179344

Simpson's rule

S = (1.5/3)·1.179344 = 0.5897

Answer

S = 0.5897

This integral has no elementary antiderivative, which is exactly when a numerical rule is needed.

2.Numerical methods

Answer: |E| = 0.2147

Numerical solution of y′ = f(t, y): Euler's method steps along the slope at the start of each step: y_n+1 = yₙ + h·f(tₙ, yₙ). Heun's method (improved Euler) averages the slope at the start and at an Euler prediction: k₁ = f(tₙ, yₙ), k₂ = f(tₙ + h, yₙ + h·k₁), y_n+1 = yₙ + (h/2)(k₁ + k₂). The classical Runge–Kutta method (RK4) combines four slopes: y_n+1 = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄). The global error of Euler is proportional to h, Heun's to h², RK4's to h⁴.

Euler steps

y₄ = 4.02528

Exact value

y(0.8) = 4.24

Global error

|E| = |4.24 − 4.02528| = 0.2147

Graph

Answer

|E| = 0.2147

Halving h roughly halves the error of Euler's method.

3.Sequences & series

Answer: S = 4

Convergence: if the terms do not tend to 0 the series diverges (n-th term test). Σ1/kᵖ converges only for p > 1, and Σrᵏ only for |r| < 1. Compare a series with one of these when its terms behave like them for large k. Ratio test: L = lim |a_k+1/a_k|; root test: L = lim |a_k|^1/k. L < 1 converges, L > 1 diverges, L = 1 decides nothing. A power series Σc_k(x − x₀)ᵏ converges for |x − x₀| < R with R = lim |c_k/c_k+1|; an alternating series with terms decreasing to 0 converges.

Partial fractions

4/(k(k + 1)) = 4·(1/k − 1/(k + 1))

Cancel

S = 4·1 = 4

Answer

S = 4

4.Sequences & series

Answer: n = 13

Taylor polynomials: near x = a a smooth function is approximated by Pₙ(x) = f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + … + f⁽ⁿ⁾(a)(x − a)ⁿ/n!; for a = 0 this is the Maclaurin polynomial. Standard series: eˣ = Σxᵏ/k!, sin x = x − x³/3! + x⁵/5! − …, cos x = 1 − x²/2! + x⁴/4! − …, ln(1 + x) = x − x²/2 + x³/3 − …, 1/(1 − x) = Σxᵏ. Lagrange error bound: |f(x) − Pₙ(x)| ≤ M·|x − a|ⁿ⁺¹/(n + 1)!, where M is the largest value of |f⁽ⁿ⁺¹⁾| between a and x.

Bounds for increasing n

n = 12: 1.052 × 10⁻⁵ ≥ 1 × 10⁻⁵

n = 13: 1.503 × 10⁻⁶ < 1 × 10⁻⁵

Smallest degree

n = 13

Answer

n = 13

5.Matrices & determinants

Answer: det A = −100

Matrices: the determinant of a 2×2 matrix is ad − bc. For 3×3, expand along a row: each entry times the determinant of the 2×2 matrix left after removing its row and column, with alternating signs + − +. Eigenvalues λ solve det(A − λI) = 0; for 2×2 that is λ² − (trace)·λ + det = 0.

2×2 minors of the first row

M₁₁ = −9, M₁₂ = 24, M₁₃ = 1

Cofactor expansion

det A = (−2)·(−9) − 5·24 + 2·1 = −100

Answer

det A = −100

6.Matrices & determinants

Answer: λ = 3

Characteristic equation

λ² − (2)λ + (−3) = 0

Solve

λ = (2 ± √(16))/2 → λ = −1, 3

λ = 3

Answer

λ = 3

7.Differential equations

Answer: y(2) = 0.345

Second-order linear equations: for y″ + b·y′ + c·y = 0 try y = e^rt: this gives the characteristic equation r² + b·r + c = 0. Two real roots r₁ ≠ r₂ give y = A·e^r₁t + B·e^r₂t (overdamped); a double root r gives y = (A + Bt)·e^rt (critically damped); complex roots −α ± jω give y = e^−αt(A cos ωt + B sin ωt) (underdamped, oscillating). The initial values y(0) and y′(0) fix A and B. Written as y″ + 2ζω_ny′ + ω_n²y = 0, ω_n is the natural frequency and ζ the damping ratio: ζ < 1 oscillates. With a constant input, y″ + by′ + cy = c·K, the solution is the steady value K plus the free response.

Characteristic roots

(r + 2)(r + 3) = 0 → r = −2, −3

Constants from the initial values

A + B = 5, −2A − 3B = 6

A = 21, B = −16

Evaluate

y(2) = 21·e⁻⁴ + (−16)·e⁻⁶ = 0.345

Answer

y(2) = 0.345

8.Differential equations

Answer: y(3) = 3.333

First-order differential equations: dy/dt = k·y has the solution y = y₀·e^kt (growth for k > 0, decay for k < 0); a half-life T gives k = ln 2/T. Newton's law of cooling, dT/dt = −k·(T − Tₐ), gives T(t) = Tₐ + (T₀ − Tₐ)·e^−kt. A linear equation y′ + p(x)·y = q(x) is solved with the integrating factor μ = e^{∫p dx}: (μy)′ = μq, so μy = ∫μq dx + C.

Integrating factor

μ = e^{∫2/x dx} = x²

Integrate

(x²y)′ = 3x² → x²y = 3x³/3 + C

Initial condition

4 = 3/3 + C → C = 3

Evaluate

y(3) = (3·3³/3 + 3)/3² = 3.333

Answer

y(3) = 3.333

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