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 ∫₀¹ 1/(1 + x³) dx with Simpson's rule using n = 2 strips. Give S.

f(x) = 1/(1 + x³)[a, b] = [0, 1]n = 2

S =

2.Numerical methodsHard

Use Euler's method with h = 0.1 for y′ = −y, y(0) = 1 to estimate y₅ ≈ y(0.5).

y′ = −yy(0) = 1h = 0.1

y₅ =

3.Sequences & seriesMedium

Find the sum S = Σ_k=1^∞ 12/(k(k + 1)).

a_k = 12/(k(k + 1))

S =

4.Sequences & seriesHard

Use the Maclaurin polynomial of degree 2 of f(x) = e^−x to approximate f(−1.5). Give P₂(−1.5).

f(x) = e^−xn = 2x = −1.5

P₂(−1.5) =

5.Matrices & determinantsMedium

Let C = AB for the matrices shown. Find the entry c₁₁ (row 1, column 1).

A = [−5 −5; 1 0]B = [5 3; 3 0]

c₁₁ =

6.Matrices & determinantsHard

Solve the system 6x − 5y = 11 and −6x − 2y = 8. Find x.

6x − 5y = 11−6x − 2y = 8

x =

7.Differential equationsMedium

A forged steel part at 340 °C is left to cool in a workshop at 23 °C. After 38 min its temperature is 171 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 90 °C?

T₀ = 340 °CT_a = 23 °CT(38) = 171 °CT = 90 °C

t = min

8.Differential equationsHard

Solve the initial value problem y′ + y/x = 5 for x > 0, with y(1) = 0. Find y(2).

y′ + y/x = 5y(1) = 0

y(2) =

3Solutions

1.Numerical methods

Answer: S = 0.8426

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 = 0.5

Σ wᵢfᵢ = 1 + 4·0.888889 + 0.5 = 5.055556

Simpson's rule

S = (0.5/3)·5.055556 = 0.8426

Answer

S = 0.8426

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

2.Numerical methods

Answer: y₅ = 0.5905

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

y₁ = 1 + 0.1·(−1) = 0.9

y₂ = 0.9 + 0.1·(−0.9) = 0.81

y₃ = 0.81 + 0.1·(−0.81) = 0.729

y₄ = 0.729 + 0.1·(−0.729) = 0.6561

y₅ = 0.6561 + 0.1·(−0.6561) = 0.59049

Graph

y₅ = 0.5905

Answer

y₅ = 0.5905

The exact value is y(0.5) = 0.606531.

3.Sequences & series

Answer: S = 12

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

12/(k(k + 1)) = 12·(1/k − 1/(k + 1))

Cancel

S = 12·1 = 12

Answer

S = 12

4.Sequences & series

Answer: P₂(−1.5) = 3.625

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

P₂(x) = 1 − x + 0.5x²

Substitute

P₂(−1.5) = 1 + 1.5 + 1.125 = 3.625

Answer

P₂(−1.5) = 3.625

The exact value is f(−1.5) = 4.481689; the error 0.856689 shrinks when you take more terms or a point closer to 0.

5.Matrices & determinants

Answer: c₁₁ = −40

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.

Row times column

c₁₁ = (−5)·5 + (−5)·3 = −40

Answer

c₁₁ = −40

6.Matrices & determinants

Answer: x = −0.4286

Determinant of the coefficient matrix

det A = 6·(−2) − (−5)·(−6) = −42

Replace the x column

det Aₓ = 11·(−2) − (−5)·8 = 18

x = 18/(−42) = −0.4286

Answer

x = −0.4286

y = −2.714

7.Differential equations

Answer: t = 77.54 min

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.

Cooling constant

k = ln(317/148)/38 = 0.02004 min⁻¹

Time to reach the target

t = ln(317/67)/0.02004 = 77.54 min

Answer

t = 77.54 min

8.Differential equations

Answer: y(2) = 3.75

Integrating factor

μ = e^{∫1/x dx} = x¹

Integrate

(x¹y)′ = 5x → x¹y = 5x²/2 + C

Initial condition

0 = 5/2 + C → C = −2.5

Evaluate

y(2) = (5·2²/2 + (−2.5))/2¹ = 3.75

Answer

y(2) = 3.75

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