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

A vehicle's speed is measured every 10 s (table). Estimate the distance s travelled from t = 0 to t = 60 s with Simpson's rule.

h = 10 sn = 6

s = m

2.Numerical methodsHard

Use Heun's method (improved Euler) with h = 0.2 for y′ = t − y, y(0) = 3 to estimate y₂ ≈ y(0.4).

y′ = t − yy(0) = 3h = 0.2

y₂ =

3.Sequences & seriesMedium

Find the radius of convergence R of the power series Σ_k=1^∞ 16ᵏx²ᵏ/k.

Σ 16ᵏx²ᵏ/kx₀ = 0

R =

4.Sequences & seriesHard

Use the Maclaurin polynomial of degree 3 of f(x) = e^−x to approximate f(1). Give P₃(1).

f(x) = e^−xn = 3x = 1

P₃(1) =

5.Matrices & determinantsMedium

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

A = [−5 2 −3; −4 −4 3; −5 3 5]

det A =

6.Matrices & determinantsHard

Find the larger eigenvalue λ of the matrix A shown.

A = [0 −3; −5 2]

λ =

7.Differential equationsMedium

A forged steel part at 380 °C is left to cool in a workshop at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 12 min.

T₀ = 380 °CT_a = 23 °Ck = 0.05 min⁻¹t = 12 min

T = °C

8.Differential equationsHard

Solve y′ + y = 2e^−0.5t with y(0) = 4. Find y(1.5).

y′ + y = 2e^−0.5ty(0) = 4

y(1.5) =

3Solutions

1.Numerical methods

Answer: s = 836.7 m

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.

Weighted sum

10 + 4·16 + 2·14 + 4·12 + 2·12 + 4·15 + 17 = 251

Simpson's rule

s = (10/3)·251 = 836.7 m

Answer

s = 836.7 m

2.Numerical methods

Answer: y₂ = 2.09

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.

Heun steps

y₁ = 3 + (0.2/2)(−3 − 2.2) = 2.48

y₂ = 2.48 + (0.2/2)(−2.28 − 1.624) = 2.0896

Graph

y₂ = 2.09

Answer

y₂ = 2.09

The exact value is y(0.4) = 2.08128.

3.Sequences & series

Answer: R = 0.25

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.

Coefficients

The series is in x²: it converges for 16·x² < 1, so |x| < 1/4.

Radius

R = 0.25

Answer

R = 0.25

The ratio test says nothing at the two end points; they have to be checked separately.

4.Sequences & series

Answer: P₃(1) = 0.3333

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.

Maclaurin polynomial

P₃(x) = 1 − x + 0.5x² − 0.1667x³

Substitute

P₃(1) = 1 − 1 + 0.5 − 0.166667 = 0.3333

Answer

P₃(1) = 0.3333

The exact value is f(1) = 0.367879; the error 0.0345461 shrinks when you take more terms or a point closer to 0.

5.Matrices & determinants

Answer: det A = 251

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 = −29, M₁₂ = −5, M₁₃ = −32

Cofactor expansion

det A = (−5)·(−29) − 2·(−5) + (−3)·(−32) = 251

Answer

det A = 251

6.Matrices & determinants

Answer: λ = 5

Characteristic equation

λ² − (2)λ + (−15) = 0

Solve

λ = (2 ± √(64))/2 → λ = −3, 5

λ = 5

Answer

λ = 5

7.Differential equations

Answer: T = 218.9 °C

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.

Solution of the cooling law

T(t) = 23 + 357·e^−0.05t

Substitute t

T(12) = 23 + 357·0.5488 = 218.9 °C

Answer

T = 218.9 °C

8.Differential equations

Answer: y(1.5) = 1.889

Integrating factor

μ = e^t

(e^ty)′ = 2e^0.5t

General solution

y = 2/0.5·e^−0.5t + C·e^−t = 4·e^−0.5t + C·e^−t

Initial condition

4 = 4 + C → C = 0

Evaluate

y(1.5) = 4·0.4724 + 0·0.2231 = 1.889

Answer

y(1.5) = 1.889

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