Numerical Analysis
Calculus and numerical methods form the computational backbone of applied mathematics. Calculus provides exact analytical tools — limits, derivatives, integrals, series — while numerical methods provide approximate algorithms for problems where exact solutions are unavailable or impractical. This skill covers both the analytical foundations and the computational techniques, including the pitfalls that arise when mathematics meets finite-precision arithmetic.
Agent affinity: euler (series, differential equations, numerical methods)
Concept IDs: math-computational-fluency, math-functions, math-expected-value, math-descriptive-statistics
Part I — Limits
The Epsilon-Delta Definition
A function f(x) has limit L as x approaches a (written lim_{x->a} f(x) = L) if for every epsilon > 0, there exists delta > 0 such that whenever 0 < |x - a| < delta, we have |f(x) - L| < epsilon.
This definition — due to Weierstrass, formalizing Cauchy's intuition — is the foundation of rigorous analysis. Every subsequent concept in calculus (continuity, derivatives, integrals) is built on limits.
Worked example. Prove that lim_{x->3} (2x + 1) = 7.
Proof. Let epsilon > 0 be given. We need |f(x) - 7| < epsilon whenever 0 < |x - 3| < delta. Compute: |(2x + 1) - 7| = |2x - 6| = 2|x - 3|. Choose delta = epsilon/2. Then 0 < |x - 3| < delta implies |f(x) - 7| = 2|x - 3| < 2(epsilon/2) = epsilon.
Limit Laws
For lim_{x->a} f(x) = L and lim_{x->a} g(x) = M:
- Sum: lim(f + g) = L + M
- Product: lim(f * g) = L * M
- Quotient: lim(f / g) = L / M (when M != 0)
- Composition: lim f(g(x)) = f(M) if f is continuous at M
L'Hopital's Rule
If lim_{x->a} f(x)/g(x) yields an indeterminate form (0/0 or infinity/infinity), and the limit of f'(x)/g'(x) exists, then:
lim_{x->a} f(x)/g(x) = lim_{x->a} f'(x)/g'(x)
Worked example. Compute lim_{x->0} sin(x)/x.
Direct substitution gives 0/0. Apply L'Hopital: lim_{x->0} cos(x)/1 = cos(0) = 1.
When NOT to use. L'Hopital requires an indeterminate form. Applying it to a determinate form (e.g., a nonzero numerator with zero denominator) gives wrong results.
The Squeeze Theorem
If g(x) <= f(x) <= h(x) near a, and lim g(x) = lim h(x) = L, then lim f(x) = L.
Worked example. Prove lim_{x->0} x^2 * sin(1/x) = 0.
Since -1 <= sin(1/x) <= 1, we have -x^2 <= x^2 sin(1/x) <= x^2. Both -x^2 and x^2 approach 0, so by the squeeze theorem, the limit is 0.
Part II — Derivatives
Definition and Rules
The derivative of f at x = a is f'(a) = lim_{h->0} (f(a + h) - f(a)) / h, when this limit exists. It measures the instantaneous rate of change.
Differentiation rules:
| Rule | Formula |
|---|
| Power | d/dx [x^n] = n*x^(n-1) |
| Sum | (f + g)' = f' + g' |
| Product | (f*g)' = f'g + fg' |
| Quotient | (f/g)' = (f'g - fg') / g^2 |
| Chain | d/dx [f(g(x))] = f'(g(x)) * g'(x) |
| Exponential | d/dx [e^x] = e^x |
| Logarithm | d/dx [ln(x)] = 1/x |
| Trig | d/dx [sin(x)] = cos(x), d/dx [cos(x)] = -sin(x) |
Applications of Derivatives
Optimization. Find critical points where f'(x) = 0 or f'(x) is undefined. Use the second derivative test: f''(c) > 0 means local minimum, f''(c) < 0 means local maximum, f''(c) = 0 is inconclusive.
Worked example. Find the dimensions of a rectangle with perimeter 20 that maximizes area.
Let the sides be x and y with 2x + 2y = 20, so y = 10 - x. Area A = x(10 - x) = 10x - x^2. Then A'(x) = 10 - 2x = 0 gives x = 5, y = 5. A''(x) = -2 < 0, confirming a maximum. The optimal rectangle is a 5 x 5 square with area 25.
Related rates. When two or more quantities change with time, differentiate their relationship implicitly with respect to time.
Mean Value Theorem. If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) with f'(c) = (f(b) - f(a)) / (b - a). This theorem is the foundation for many proofs in analysis.
Part III — Integration
Riemann Sums and the Definite Integral
The definite integral from a to b of f(x)dx is defined as the limit of Riemann sums:
integral_a^b f(x)dx = lim_{n->infinity} sum_{i=1}^{n} f(x_i*) * Delta_x
where Delta_x = (b - a)/n and x_i* is a sample point in the i-th subinterval.
The Fundamental Theorem of Calculus
Part 1 (FTC1). If f is continuous on [a, b], then F(x) = integral_a^x f(t)dt is differentiable and F'(x) = f(x). Integration and differentiation are inverse operations.
Part 2 (FTC2). If F is any antiderivative of f on [a, b], then integral_a^b f(x)dx = F(b) - F(a).
Integration Techniques
Substitution (u-substitution). For integral f(g(x)) * g'(x) dx, let u = g(x), du = g'(x)dx.
Worked example. Compute integral 2x * cos(x^2) dx.
Let u = x^2, du = 2x dx. The integral becomes integral cos(u) du = sin(u) + C = sin(x^2) + C.
Integration by parts. integral u dv = uv - integral v du. Choose u and dv using the LIATE heuristic: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential — u should be the type appearing earliest in this list.
Worked example. Compute integral x * e^x dx.
Let u = x (algebraic), dv = e^x dx. Then du = dx, v = e^x. By parts: xe^x - integral e^x dx = xe^x - e^x + C = e^x(x - 1) + C.
Partial fractions. Decompose a rational function into simpler fractions before integrating. Factor the denominator, then express as a sum of fractions with linear and irreducible quadratic denominators.
Trigonometric substitution. For integrands involving sqrt(a^2 - x^2), sqrt(a^2 + x^2), or sqrt(x^2 - a^2), substitute x = asin(theta), x = atan(theta), or x = a*sec(theta) respectively.
Part IV — Series
Convergence Tests
A series sum_{n=1}^{infinity} a_n converges if the sequence of partial sums S_N = sum_{n=1}^{N} a_n has a finite limit.
| Test | Statement | Use when |
|---|
| Divergence | If lim a_n != 0, the series diverges | Quick first check |
| Geometric | sum r^n converges iff | r |
| p-series | sum 1/n^p converges iff p > 1 | Power-law terms |
| Integral | sum a_n and integral f(x)dx converge/diverge together | f is positive, decreasing, continuous |
| Comparison | 0 <= a_n <= b_n and sum b_n converges => sum a_n converges | Can bound by a known series |
| Limit comparison | lim a_n/b_n = L (0 < L < infinity) => same behavior | Asymptotic comparison |
| Ratio | lim | a_{n+1}/a_n |
| Root | lim | a_n |
| Alternating series | sum (-1)^n b_n converges if b_n decreases to 0 | Alternating sign |
Taylor and Maclaurin Series
The Taylor series of f about x = a is:
f(x) = sum_{n=0}^{infinity} f^(n)(a) / n! * (x - a)^n
The Maclaurin series is the Taylor series about a = 0.
Key Maclaurin series:
| Function | Series | Radius of convergence |
|---|
| e^x | sum x^n / n! | infinity |
| sin(x) | sum (-1)^n x^(2n+1) / (2n+1)! | infinity |
| cos(x) | sum (-1)^n x^(2n) / (2n)! | infinity |
| 1/(1-x) | sum x^n | 1 |
| ln(1+x) | sum (-1)^(n+1) x^n / n | 1 |
| arctan(x) | sum (-1)^n x^(2n+1) / (2n+1) | 1 |
Taylor's remainder theorem. If f has n+1 continuous derivatives on an interval containing a and x, then:
f(x) = P_n(x) + R_n(x)
where P_n is the n-th degree Taylor polynomial and |R_n(x)| <= M * |x - a|^(n+1) / (n+1)! for M = max |f^(n+1)(t)| on the interval. This bounds the approximation error.
Part V — Differential Equations
Separable Equations
Form: dy/dx = f(x) * g(y). Separate: dy/g(y) = f(x) dx. Integrate both sides.
Worked example. Solve dy/dx = xy, y(0) = 1.
Separate: dy/y = x dx. Integrate: ln|y| = x^2/2 + C. Exponentiate: y = A * e^(x^2/2). Initial condition y(0) = 1 gives A = 1. Solution: y = e^(x^2/2).
First-Order Linear Equations
Form: dy/dx + P(x)*y = Q(x). Multiply by integrating factor mu(x) = e^(integral P(x) dx):
d/dx [mu(x) * y] = mu(x) * Q(x)
Integrate both sides to solve.
Worked example. Solve dy/dx + 2y = e^(-x), y(0) = 3.
Integrating factor: mu = e^(2x). Multiply: d/dx [e^(2x) y] = e^(2x) * e^(-x) = e^x. Integrate: e^(2x) y = e^x + C. So y = e^(-x) + C * e^(-2x). Initial condition: 3 = 1 + C, so C = 2. Solution: y = e^(-x) + 2e^(-2x).
Exact Equations
Form: M(x,y) dx + N(x,y) dy = 0 is exact if dM/dy = dN/dx. Then there exists F(x,y) with dF/dx = M and dF/dy = N. The solution is F(x,y) = C.
Second-Order Linear with Constant Coefficients
Form: ay'' + by' + cy = 0. The characteristic equation ar^2 + br + c = 0 yields:
- Two distinct real roots r_1, r_2: y = C_1 e^(r_1 x) + C_2 e^(r_2 x)
- Repeated root r: y = (C_1 + C_2 x) e^(rx)
- Complex roots alpha +/- beta*i: y = e^(alpha x) (C_1 cos(beta x) + C_2 sin(beta x))
Part VI — Numerical Methods
Newton-Raphson Method
To find a root of f(x) = 0, iterate: x_{n+1} = x_n - f(x_n) / f'(x_n).
Convergence: Quadratic when it converges (errors square each iteration). Requires f'(x_n) != 0 and a sufficiently good initial guess.
Worked example. Approximate sqrt(2) as a root of f(x) = x^2 - 2.
f'(x) = 2x. Starting at x_0 = 1:
x_1 = 1 - (1 - 2)/(2) = 1.5
x_2 = 1.5 - (2.25 - 2)/(3) = 1.5 - 0.0833... = 1.41667
x_3 = 1.41667 - (2.00694 - 2)/(2.83333) = 1.41422
After 3 iterations, we have 5 correct digits.
Failure modes. Diverges when the initial guess is near a stationary point (f' near 0), near an inflection point, or when the function oscillates. Always verify convergence.
Euler's Method
For dy/dx = f(x, y) with y(x_0) = y_0, approximate: y_{n+1} = y_n + h * f(x_n, y_n), where h is the step size.
Error. Global error is O(h) — first-order method. Halving the step size halves the error but doubles the work.
Improved Euler (Heun's method). Use the average of slopes at the start and end of each step. Global error is O(h^2).
Runge-Kutta (RK4). The workhorse of ODE solvers. Fourth-order accuracy (global error O(h^4)) with four function evaluations per step. The standard formulas:
k_1 = h * f(x_n, y_n)
k_2 = h * f(x_n + h/2, y_n + k_1/2)
k_3 = h * f(x_n + h/2, y_n + k_2/2)
k_4 = h * f(x_n + h, y_n + k_3)
y_{n+1} = y_n + (k_1 + 2k_2 + 2k_3 + k_4) / 6
Simpson's Rule
For numerical integration: integral_a^b f(x) dx approximately equals (h/3) [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{n-1}) + f(x_n)], where h = (b - a)/n and n is even.
Error bound. |E_S| <= M * (b - a)^5 / (180 * n^4), where M = max |f^(4)(x)| on [a, b]. Fourth-order accuracy — far superior to the trapezoidal rule (O(n^2)) for smooth functions.
Part VII — Computational Pitfalls
Floating-Point Arithmetic
IEEE 754 double-precision floating point provides about 15-16 significant decimal digits. Every floating-point operation introduces a rounding error of relative magnitude at most epsilon_machine approximately 2.2e-16.
Catastrophic Cancellation
When two nearly equal numbers are subtracted, leading significant digits cancel and the result retains only the trailing (noisy) digits. Example: computing f(x) = (1 - cos(x)) / x^2 near x = 0 suffers catastrophic cancellation because both 1 and cos(x) are near 1.
Fix: Use the identity 1 - cos(x) = 2sin^2(x/2), so f(x) = 2sin^2(x/2)/x^2 = (sin(x/2)/(x/2))^2 / 2, which is numerically stable.
Numerical Stability
An algorithm is numerically stable if small perturbations in the input produce small perturbations in the output. An unstable algorithm amplifies errors.
Example. Forward recurrence for the integral I_n = integral_0^1 x^n * e^(x-1) dx using I_n = 1 - n*I_{n-1} is unstable (errors grow by factor n). Backward recurrence from a large N down is stable.
Conditioning
The condition number kappa(A) of a matrix A measures how sensitive Ax = b is to perturbations in A or b. For linear systems: relative error in x <= kappa(A) * relative error in b. When kappa(A) is large, the problem is ill-conditioned and even a stable algorithm produces inaccurate results.
Coprocessor Acceleration
This skill declares coprocessor: [algebrus, statos] in its frontmatter — making it the first shipped skill to declare a consumer of the math coprocessor (coprocessors/math/, the coprocessor chipset's canonical example). The declaration is read by the activation pipeline's pre-warm stage createCoprocessorStage (→ extractCoprocessorRaw → parseCoprocessorSpec → activateCoprocessor, in src/coprocessor/), which — when the coprocessor is enabled (default-on; disable it with { "gsd-skill-creator": { "coprocessor": { "enabled": false } } } in .claude/gsd-skill-creator.json) and a Python MCP server with mcp + numpy + scipy is reachable — pre-warms the named chips so the first real tool call doesn't pay the discovery cost. Note: that stage runs only when a host drives skills through SkillApplicator.apply(); the shipped skill-creator invoke CLI loads skills directly and does not yet run it, so today the pre-warm is exercised by the library apply() path and the tests below, not by a default CLI session.
The declared chips back this skill's computational content:
algebrus (linear algebra) — gemm (GPU-accelerated via cuBLAS whenever a CUDA device is present), solve/svd (GPU-promoted when cuSOLVER is available), and det (CPU oracle) underpin the linear systems behind Newton's method for systems, least-squares fits, and the conditioning analysis above. Where a GPU path runs, the NumPy/SciPy CPU oracle returns the same result up to floating-point tolerance; meta.device reports which path actually ran.statos (statistics) — describe, regression, and monte_carlo cover descriptive statistics, polynomial regression, and stochastic integration (monte_carlo is the cuRAND-accelerated path when a GPU is present).
Op-conformance note. algebrus.eigen is intentionally CPU-only — it has no GPU acceleration path by design (see the verdict in coprocessors/math/PACKAGE.md). A separate, deferred wire-serialization defect makes eigen unusable through the typed client today (documented alongside that verdict); gemm/solve/svd/det are the wire-safe surface this skill builds on.
The end-to-end activation path against a live server is proven in src/coprocessor/__tests__/numerical-analysis-coprocessor.integration.test.ts (gated on COPROCESSOR_LIVE_TESTS=1); the always-on round-trip of this shipped frontmatter is proven in numerical-analysis-coprocessor.test.ts.
When to Use This Skill
- Computing limits, derivatives, or integrals (exact or numerical)
- Solving differential equations (exact or numerical)
- Analyzing series convergence and computing Taylor approximations
- Performing optimization (finding maxima/minima)
- Assessing numerical accuracy and stability of computations
When NOT to Use This Skill
- For algebraic equation solving without calculus — use algebraic-reasoning
- For geometric reasoning — use geometric-intuition
- For conjecture generation from patterns — use pattern-recognition
- For real-world problem formulation — use mathematical-modeling (though this skill handles the computational engine once the model is formulated)
Cross-References
- euler agent: Differential equations, series, numerical methods, and connections between analysis and computation. Named for Leonhard Euler, the most prolific mathematician in history, who contributed foundationally to calculus, series theory, and numerical methods.
- algebraic-reasoning skill: Algebraic manipulation used in integration, differential equations, and series computation.
- geometric-intuition skill: Geometric interpretation of derivatives (tangent lines), integrals (area under curves), and series (geometric series).
- mathematical-modeling skill: Formulating real-world problems that this skill then solves computationally.
References
- Rudin, W. (1976). Principles of Mathematical Analysis. 3rd edition. McGraw-Hill.
- Spivak, M. (2008). Calculus. 4th edition. Publish or Perish.
- Stewart, J. (2015). Calculus: Early Transcendentals. 8th edition. Cengage.
- Burden, R. L., & Faires, J. D. (2010). Numerical Analysis. 9th edition. Brooks/Cole.
- Trefethen, L. N., & Bau, D. (1997). Numerical Linear Algebra. SIAM.
- Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms. 2nd edition. SIAM.
- Goldberg, D. (1991). "What every computer scientist should know about floating-point arithmetic." ACM Computing Surveys, 23(1), 5-48.
