Numerical Methods, Tolerances, and Reproducibility

This reference targets MATLAB R2026a. Confirm every non-base product before using toolbox-specific functions.

Linear systems and decompositions

Solve systems; do not form an inverse as an intermediate:

x = A \ b;
residual = A*x - b;
relativeResidual = norm(residual) / ...
    max(norm(A)*norm(x) + norm(b), realmin(class(A)));

Check dimensions, rank/conditioning, scaling, symmetry, definiteness, and sparsity. A small residual does not guarantee a small forward error for an ill-conditioned problem.

Common base MATLAB operations include:

• lu, qr, chol, ldl, schur;
• eig, svd, eigs, svds;
• rank, cond, rcond, norm, pinv;
• lsqminnorm, lsqnonneg, and backslash least squares.

Use an economy decomposition where appropriate and request only the spectrum needed for large/sparse problems. Eigenvector signs/phases and bases in degenerate subspaces are not unique; compare invariant quantities rather than raw vectors.

Floating-point comparison

Binary floating point does not represent most decimal fractions exactly. Choose tolerances from the model, scale, conditioning, discretization, measurement uncertainty, and algorithm—not from a universal constant.

A robust scalar/elementwise policy often has the form:

errorMagnitude = abs(actual - expected);
limit = absoluteTolerance + relativeTolerance .* abs(expected);
isAcceptable = errorMagnitude <= limit;

Handle these explicitly:

• expected values near zero need an absolute tolerance;
• large expected values often need a relative tolerance;
• NaN equality is a semantic decision (isequaln differs from ==);
• Inf signs should match when infinity is expected;
• class, size, sparsity, and complex values are part of the contract.

R2026a documents isapprox alongside equality operations. In matlab.unittest, use AbsTol/RelTol or AbsoluteTolerance/RelativeTolerance. Record why values are scientifically acceptable.

Do not widen tolerances automatically after an upgrade. First investigate RNG, ordering, reduction order, solver defaults/options, data type, threading, compiler, library, and release-note changes.

Random streams

Record algorithm and seed, not only a seed:

rng(1729, "twister");
stateAtStart = rng;
samples = randn(1000, 1);

For local independent streams:

stream = RandStream("Threefry", Seed=1729);
stream.Substream = 4;
samples = randn(stream, 1000, 1);

Generator availability and bitwise sequences can vary by algorithm/release. Avoid rng("shuffle") for reproducible work. On parallel workers, time-based seeding can collide; use supported independent streams/substreams and record worker mapping. Parallel computing requires Parallel Computing Toolbox.

Integration, roots, and differential equations

Base MATLAB provides general numerical methods including:

• integral, integral2, integral3, trapz, cumtrapz;
• gradient, diff;
• fzero;
• ODE solvers such as ode45, ode23, ode113, ode15s, ode23s, ode23t, and ode23tb;
• boundary-value solvers such as bvp4c and bvp5c.

Define tolerances and failure criteria:

options = odeset( ...
    RelTol=1e-7, ...
    AbsTol=1e-10, ...
    MaxStep=0.05);
[t, y] = ode45(@rhs, [0 5], 1, options);

Solver tolerances control local error estimates, not proof of a globally correct model. Check conservation laws, event localization, stiffness, step-size convergence, and an independent formulation. R2026a adds an automatic-differentiation Jacobian option for the ode object; verify the specific solver/problem and release notes before using it. R2026b extends ode automatic Jacobians to complex equations and adds the Waypoints property for time values the solver must reach exactly. These are release-specific ode object features, not ode45 options.

Optimization and fitting boundaries

Base MATLAB includes fminsearch and fminbnd. These do not replace constrained or specialized solvers.

Examples of separately licensed boundaries:

Some base functions have similarly named toolbox alternatives. Check the function's current product page and the project dependency report; never infer ownership from a code example.

Optimization reproducibility requires objective/constraint definitions, starting points, bounds, solver/options, stopping tolerances, gradients, scaling, RNG state for stochastic methods, and exit diagnostics. Compare feasibility and optimality measures, not only the objective value.

Statistics and signal processing

Base array summaries include mean, median, std, var, min, max, movmean, movmedian, cov, corrcoef, histcounts, and polynomial polyfit/polyval. Some distribution, model, hypothesis-test, robust, classification, and specialized plotting APIs require Statistics and Machine Learning Toolbox.

For FFT work:

n = numel(x);
Y = fft(x);
frequency = (0:n-1).' * (sampleRate/n);

Document sample rate, units, window, detrending, normalization, one- versus two-sided spectrum, zero padding, and endpoint convention. fft and conv are base MATLAB; many filter-design and spectral-estimation functions are Signal Processing Toolbox.

Verification patterns

Use several layers:

1. Dimensional/invariant checks: sizes, units, conservation, monotonicity, positivity, symmetry.
2. Analytic cases: small problems with known solutions.
3. Refinement studies: mesh, step, quadrature, or tolerance convergence.
4. Independent implementation: alternative solver or formulation.
5. Condition/sensitivity analysis: perturb inputs and options.
6. Release comparison: compare scientifically meaningful observables with a documented tolerance.
7. Performance measurement: after correctness, measure representative workloads with timeit.

Do not claim bitwise reproducibility across releases, hardware, thread counts, GPU/CPU, or external libraries unless it was actually tested and documented.

Reproducibility record

At minimum capture:

• MATLAB release/update or Octave version;
• OS and architecture, only as named fields;
• required products and license status separately;
• source/input hashes and schema versions;
• numeric classes and shapes;
• RNG algorithm, seed, substream, and parallel mapping;
• solver names/options/tolerances and stopping diagnostics;
• expected invariants and acceptance tolerances;
• output format/version and graphics export settings.

Use scripts/reproducibility_report.py to hash only named local artifacts. It does not inspect the broad environment.

Sources (reviewed 2026-10-01)

• Linear Algebra
• [mldivide](https://www.mathworks.com/help/matlab/ref/double.mldivide.html)
• [eq floating-point guidance and isapprox](https://www.mathworks.com/help/matlab/ref/double.eq.html)
• [AbsoluteTolerance](https://www.mathworks.com/help/matlab/ref/matlab.unittest.constraints.absolutetolerance-class.html)
• [RelativeTolerance](https://www.mathworks.com/help/matlab/ref/matlab.unittest.constraints.relativetolerance-class.html)
• [rng](https://www.mathworks.com/help/matlab/ref/rng.html)
• [RandStream](https://www.mathworks.com/help/matlab/ref/randstream.html)
• ODE Solvers
• [ode properties and version history](https://www.mathworks.com/help/matlab/ref/ode.html)
• Optimization
• MATLAB product list and pricing/licensing
• MATLAB release notes