Math

Overview

Mathematical computation forms the backbone of quantitative analysis in science, engineering, finance, and data science. This comprehensive library of mathematical functions extends Excel’s capabilities far beyond native formulas, providing access to sophisticated numerical methods implemented in SciPy, NumPy, CasADi, and specialized Python libraries.

Modern mathematical computing requires far more than basic arithmetic—it demands robust algorithms for solving differential equations, fitting complex models to data, optimizing multivariate functions, and performing matrix operations with numerical stability. These functions bridge the gap between Excel’s spreadsheet interface and the power of production-grade scientific computing libraries, enabling researchers and engineers to tackle challenging mathematical problems directly from within a familiar spreadsheet environment.

Calculus and Differential Equations form the foundation for modeling dynamic systems. Whether computing derivatives of multivariable functions through differentiation, evaluating integrals numerically, or solving systems of ordinary differential equations, these tools enable accurate analysis of continuous change. Applications range from sensitivity analysis in optimization to pharmacokinetic modeling in drug development and epidemic modeling in public health.

Curve Fitting and Data Approximation allow you to discover the relationships hidden within data. Using least-squares regression, you can fit symbolic models or pre-defined curves to experimental measurements. This is essential across fields: chemists use adsorption and binding models to understand molecular interactions; biologists apply dose-response and enzyme kinetics models; engineers work with rheological and spectroscopic models. Advanced methods like CasADi-based fitting provide automatic differentiation for complex symbolic expressions, while spline-based interpolation reconstructs smooth curves from scattered or structured data.

Interpolation and Approximation bridge discrete data points with smooth continuous functions. Methods range from simple univariate splines and polynomial interpolation to sophisticated multidimensional approaches like radial basis functions. These are invaluable when you need to evaluate functions at arbitrary points, smooth noisy measurements, or reconstruct surfaces from scattered measurements.

Linear Algebra provides the computational foundation for modern numerical analysis. From matrix decompositions (QR, SVD, Cholesky) to solving linear systems and pseudoinverse computations, these operations underpin virtually every scientific computation. They are essential for solving overdetermined and underdetermined systems, analyzing system stability through eigenvalue problems, and transforming data for machine learning applications.

Optimization addresses the central computational challenge: finding the best solution within constraints. Assignment problems like the Hungarian algorithm match resources optimally. Local methods converge rapidly near a solution, suitable when good starting points are available. Global methods like differential evolution and dual annealing explore broader solution spaces to escape local minima. Linear and quadratic programming handle structured problems efficiently, while root-finding tools solve systems of nonlinear equations that arise throughout engineering and science. The reference figure illustrates these key mathematical domains and their interconnections.

Figure 1: Mathematical Computing Landscape: (A) Core mathematical domains and their primary applications. (B) Optimization landscape showing local minima and global structure.

Calculus

Differentiation

Tool	Description
HESSIAN	Compute the Hessian matrix (second derivatives) of a scalar function using CasADi symbolic differentiation.
JACOBIAN	Calculate the Jacobian matrix of mathematical expressions with respect to specified variables.
SENSITIVITY	Compute the sensitivity of a scalar model with respect to its parameters using CasADi.

Integration

Tool	Description
DBLQUAD	Compute the double integral of a function over a two-dimensional region.
QUAD	Numerically integrate a function defined by a table of x, y values over [a, b] using adaptive quadrature.
TRAPEZOID	Integrate sampled data using the composite trapezoidal rule.

Ode Models

Tool	Description
BRUSSELATOR	Numerically solves the Brusselator system of ordinary differential equations for autocatalytic chemical reactions.
COMPARTMENTAL_PK	Numerically solves the basic one-compartment pharmacokinetics ODE using scipy.integrate.solve_ivp.
FITZHUGH_NAGUMO	Numerically solves the FitzHugh-Nagumo system of ordinary differential equations for neuron action potentials using scipy.integrate.solve_ivp.
HODGKIN_HUXLEY	Numerically solves the Hodgkin-Huxley system of ordinary differential equations for neuron action potentials.
LORENZ	Numerically solves the Lorenz system of ordinary differential equations for chaotic dynamics.
LOTKA_VOLTERRA	Numerically solves the Lotka-Volterra predator-prey system of ordinary differential equations.
MICHAELIS_MENTEN	Numerically solves the Michaelis-Menten system of ordinary differential equations for enzyme kinetics using scipy.integrate.solve_ivp.
SEIR	Numerically solves the SEIR system of ordinary differential equations for infectious disease modeling using scipy.integrate.solve_ivp.
SIR	Solves the SIR system of ordinary differential equations for infection dynamics using scipy.integrate.solve_ivp (see scipy.integrate.solve_ivp).
VAN_DER_POL	Numerically solves the Van der Pol oscillator system of ordinary differential equations.

Ode Systems

Tool	Description
SOLVE_BVP	Solve a boundary value problem for a second-order system of ODEs.
SOLVE_IVP	Solve an initial value problem for a system of ODEs of the form dy/dt = A @ y.

Curve Fitting

Least Squares

Tool	Description
CA_CURVE_FIT	Fit an arbitrary symbolic model to data using CasADi and automatic differentiation.
CURVE_FIT	Fit a model expression to xdata, ydata using scipy.optimize.curve_fit.
LM_FIT	Fit data using lmfit’s built-in models with optional model composition.
MINUIT_FIT	Fit an arbitrary model expression to data using iminuit least-squares minimization with uncertainty estimates.

Models

Tool	Description
ADSORPTION	Fits adsorption models to data using scipy.optimize.curve_fit.
AGRICULTURE	Fits agriculture models to data using scipy.optimize.curve_fit.
BINDING_MODEL	Fits binding_model models to data using scipy.optimize.curve_fit.
CHROMA_PEAKS	Fits chroma_peaks models to data using scipy.optimize.curve_fit.
DOSE_RESPONSE	Fits dose_response models to data using scipy.optimize.curve_fit.
ELECTRO_ION	Fits electro_ion models to data using scipy.optimize.curve_fit.
ENZYME_BASIC	Fits enzyme_basic models to data using scipy.optimize.curve_fit.
ENZYME_INHIBIT	Fits enzyme_inhibit models to data using scipy.optimize.curve_fit.
EXP_ADVANCED	Fits exp_advanced models to data using scipy.optimize.curve_fit.
EXP_DECAY	Fits exp_decay models to data using scipy.optimize.curve_fit.
EXP_GROWTH	Fits exponential growth models to data using scipy.optimize.curve_fit.
GROWTH_POWER	Fits growth_power models to data using scipy.optimize.curve_fit.
GROWTH_SIGMOID	Fits growth_sigmoid models to data using scipy.optimize.curve_fit.
MISC_PIECEWISE	Fits misc_piecewise models to data using scipy.optimize.curve_fit.
PEAK_ASYM	Fits peak_asym models to data using scipy.optimize.curve_fit.
POLY_BASIC	Fits poly_basic models to data using scipy.optimize.curve_fit.
RHEOLOGY	Fits rheology models to data using scipy.optimize.curve_fit.
SPECTRO_PEAKS	Fits spectro_peaks models to data using scipy.optimize.curve_fit.
STAT_DISTRIB	Fits stat_distrib models to data using scipy.optimize.curve_fit.
STAT_PARETO	Fits stat_pareto models to data using scipy.optimize.curve_fit.
WAVEFORM	Fits waveform models to data using scipy.optimize.curve_fit.

Interpolation

Approximation

Tool	Description
LAGRANGE_INTERP	Compute the Lagrange interpolating polynomial through a set of points.
PADE	Compute Pade rational approximation to a polynomial.

Multivariate

Tool	Description
GRID_INTERP	Interpolator on a regular grid in 2D.
GRIDDATA	Interpolate unstructured D-D data.
INTERPN	Multidimensional interpolation on regular grids (2D).
LINEAR_ND_INTERP	Piecewise linear interpolator in N > 1 dimensions.
NEAREST_ND_INTERP	Nearest neighbor interpolation in N > 1 dimensions.
RBF_INTERPOLATOR	Radial basis function interpolation in N dimensions.

Splines

Tool	Description
INTERP_UV_SPLINE	1-D interpolating spline for data.
MAKE_INTERP_SPLINE	Compute interpolating B-spline and evaluate at new points.
MAKE_LSQ_SPLINE	Compute LSQ-based fitting B-spline.
SMOOTH_SPLINE	Smoothing cubic spline.
UNIVARIATE_SPLINE	1-D smoothing spline fit to data.

Univariate

Tool	Description
AKIMA_INTERP	Akima 1D interpolation.
BARYCENTRIC_INTERP	Interpolating polynomial for a set of points using barycentric interpolation.
CUBIC_SPLINE	Cubic spline data interpolator.
HERMITE_SPLINE	Piecewise-cubic interpolator matching values and first derivatives.
INTERP1D	Interpolate a 1-D function.
KROGH_INTERPOLATE	Krogh polynomial interpolation.
PCHIP_INTERPOLATE	PCHIP 1-D monotonic cubic interpolation.

Linear Algebra

Tool	Description
CHOLESKY	Compute the Cholesky decomposition of a real, symmetric positive-definite matrix.
EXPM	Compute the matrix exponential of a square matrix using scipy.linalg.expm
LSQ_LINEAR	Solve a bounded linear least-squares problem.
LSTSQ	Compute the least-squares solution to Ax = B using scipy.linalg.lstsq.
PINV	Compute the Moore-Penrose pseudoinverse of a matrix using singular value decomposition (SVD).
QR	Compute the QR decomposition of a matrix and return either Q or R.
SVD	Compute the Singular Value Decomposition (SVD) of a matrix using scipy.linalg.svd.

Optimization

Assignment Problems

Tool	Description
LINEAR_ASSIGNMENT	Solve the linear assignment problem using scipy.optimize.linear_sum_assignment.
QUAD_ASSIGN	Solve a quadratic assignment problem using SciPy’s implementation.

Global Optimization

Tool	Description
BASIN_HOPPING	Minimize a single-variable expression with SciPy’s basinhopping algorithm.
BRUTE	Perform a brute-force grid search to approximate the global minimum of a function.
DIFF_EVOLUTION	Minimize a multivariate function using differential evolution.
DUAL_ANNEALING	Minimize a multivariate function using dual annealing.
SHGO	Find global minimum using Simplicial Homology Global Optimization.

Linear Programming

Tool	Description
CA_QUAD_PROG	Solve a quadratic programming problem using CasADi’s qpsol solver.
LINEAR_PROG	Solve a linear programming problem using SciPy’s linprog function.
MILP	Solve a mixed-integer linear program using scipy.optimize.milp.

Local Optimization

Tool	Description
CA_MINIMIZE	Minimize a multivariate function using CasADi with automatic differentiation.
MINIMIZE	Minimize a multivariate function using SciPy’s minimize routine.
MINIMIZE_SCALAR	Minimize a single-variable function using SciPy’s minimize_scalar.

Root Finding

Tool	Description
CA_ROOT	Solve a system of nonlinear equations using CasADi with automatic Jacobian.
FIXED_POINT	Find a fixed point x such that f(x) = x for a scalar function expression.
ROOT	Solve a square nonlinear system using SciPy’s root solver.
ROOT_SCALAR	Find a real root of a scalar function using SciPy’s root_scalar.