Algebra Functions (92)

Perform partial fraction decomposition on a rational expression. Decomposes p(x)/q(x) into a sum of simpler fractions when the denominator factors into distinct linear terms.

Declare an assumption about a symbolic variable. Valid properties: positive, negative, integer, real, complex, nonnegative, nonzero. Assumptions are stored globally and can be used by reduce().

Find the asymptotic (leading) behavior of an expression as a variable approaches a point. For rational functions, returns the leading term ratio. For polynomials, returns the leading monomial. For other expressions, falls back to computing the limit.

Cancel common factors in a rational expression. Simplifies a fraction by cancelling common factors between the numerator and denominator.

Extract polynomial coefficients of an expression with respect to a variable. Returns an array [c0, c1, c2, ...] where expr = c0 + c1*x + c2*x^2 + ...

Collect terms of a polynomial expression by powers of a variable. Groups like powers and combines their coefficients.

Combine like terms and apply logarithm/power product rules. Inverse of expand for patterns like log(a)+log(b)->log(a*b), a^n*a^m->a^(n+m).

Expand an expression treating the listed variables as complex numbers. Substitutes each variable x with x_re + i*x_im, expands, and separates into real and imaginary parts.

Approximate minimum degree ordering. The minimum degree algorithm is a widely used heuristic for finding a permutation P so that P*A*P' has fewer nonzeros in its factorization than A. It is a gready method that selects the sparsest pivot row and column during the course of a right looking sparse Cho

Computes the Cholesky factorization of matrix A. It computes L and P so L * L' = P * A * P'

Computes the column counts using the upper triangular part of A. It transposes A internally, none of the input parameters are modified.

Computes the numeric LU factorization of the sparse matrix A. Implements a Left-looking LU factorization algorithm that computes L and U one column at a tume. At the kth step, it access columns 1 to k-1 of L and column k of A. Given the fill-reducing column ordering q (see parameter s) computes L, U

The function csSpsolve() computes the solution to G * x = bk, where bk is the kth column of B. When lo is true, the function assumes G = L is lower triangular with the diagonal entry as the first entry in each column. When lo is true, the function assumes G = U is upper triangular with the diagonal

Symbolic ordering and analysis for QR and LU decompositions. symbolic ordering and analysis for LU decomposition (false)

Computes the symmetric permutation of matrix A accessing only the upper triangular part of A. C = P * A * P'

Compute the symbolic curl of a 3D vector field. The curl is defined as [dF3/dy - dF2/dz, dF1/dz - dF3/dx, dF2/dx - dF1/dy]. Both field and variables must have exactly 3 elements.

Find the degree of a polynomial expression in a specified variable. Walks the expression tree to find the highest power of the given variable. Returns 0 for constant expressions and -Infinity for non-polynomial expressions.

Takes the derivative of an expression expressed in parser Nodes. The derivative will be taken over the supplied variable in the second parameter. If there are multiple variables in the expression, it will return a partial derivative.

Compute finite differences of a numeric sequence. First differences give the discrete analog of the derivative: Δa[i] = a[i+1] - a[i]. Higher-order differences are computed by applying the operator repeatedly.

Compute the directional derivative of a scalar expression in the given direction. D_v f = gradient(f) · (v / |v|). The direction vector is normalized before computing the dot product with the gradient.

Compute the discriminant of a polynomial. For quadratic ax^2+bx+c: disc = b^2-4ac. For cubic: disc = 18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2. For higher degrees, uses the resultant of p and its derivative.

Compute the symbolic divergence of a vector field. The divergence is the sum of partial derivatives: div F = dF1/dx1 + dF2/dx2 + ... The field and variables arrays must have the same length.

Assert that a variable belongs to a mathematical domain and record the assumption. Valid domains: Integer, Real, Rational, Complex, Positive, Negative. Returns true when the assumption is stored.

Eliminate specified variables from a system of polynomial equations using resultants. For each variable to eliminate, computes resultants between pairs of equations to produce equations in the remaining variables.

Convert exponential expressions to trigonometric or hyperbolic form using Euler\

Expand an algebraic expression by distributing multiplication over addition and expanding integer powers. Returns the expanded form of the expression.

Factor a polynomial expression into its irreducible factors. Uses the rational roots theorem to find rational roots, then performs synthetic division. Handles linear, quadratic, cubic, and higher-degree polynomials with rational roots.

Compute the Fourier series coefficients of a periodic function using numeric integration (Simpson\

Aggressively simplify an expression by applying multiple strategies (basic simplify, trig identities, expand, rationalize) and returning the result with the fewest nodes.

Expand special function identities: gamma recurrence gamma(n+1)->n*gamma(n), beta(a,b)->gamma(a)*gamma(b)/gamma(a+b), binomial coefficients, and evaluates gamma/factorial at integer arguments.

Compute the symbolic gradient of a scalar expression with respect to a list of variables. Returns an array of strings, where each element is the partial derivative with respect to the corresponding variable.

Compute a Gröbner basis for a polynomial ideal. Supports 1-variable systems (via GCD) and 2-variable systems (via resultant-based reduction) with lexicographic ordering.

Compute the implicit derivative dy/dx of an equation F(x, y) = 0 using the formula dy/dx = -(dF/dx) / (dF/dy).

Compute the symbolic indefinite integral of an expression with respect to a variable. Handles polynomials (power rule), basic trigonometric functions (sin, cos), exponentials (exp), and 1/x. Uses linearity of integration for sums and constant multiples. Returns the integral without the constant of i

Convenience alias for inverseLaplaceTransform. Compute the inverse Laplace transform using a lookup table of known transform pairs. Supported: 1/s → 1, 1/s^2 → t, c/s^n → c*t^(n-1)/(n-1)!, 1/(s-a) → e^(at), s/(s^2+b^2) → cos(bt), b/(s^2+b^2) → sin(bt), and more.

Compute the inverse Laplace transform using a lookup table of known transform pairs. Supported patterns include: 1/s → 1, 1/s^2 → t, c/s^n → c*t^(n-1)/(n-1)!, 1/(s-a) → e^(at), s/(s^2+b^2) → cos(bt), b/(s^2+b^2) → sin(bt), s/(s^2-a^2) → cosh(at), a/(s^2-a^2) → sinh(at). Sums are handled term by term

Compute the symbolic Jacobian matrix of a vector of expressions with respect to a list of variables. J[i][j] = d(exprs[i]) / d(variables[j]). Returns a 2D array of strings.

Compute the Laplace transform of an expression using a table-based approach. Supports: constants (1 -> 1/s), powers (t^n -> n!/s^(n+1)), exponentials (e^(at) -> 1/(s-a)), sine (sin(bt) -> b/(s^2+b^2)), cosine (cos(bt) -> s/(s^2+b^2)), and linearity for sums and scalar multiples.

Compute the scalar Laplacian of an expression with respect to a list of variables. The Laplacian is the sum of second partial derivatives: Δf = ∂²f/∂x² + ∂²f/∂y² + ...

Computes the number of leaves in the parse tree of the given expression

Compute the limit of an expression as a variable approaches a value. Uses direct substitution first; for indeterminate 0/0 forms, applies L\

Finds one solution of the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector.

Finds all solutions of the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector.

Calculate the Matrix LU decomposition with partial pivoting. Matrix A is decomposed in three matrices (L, U, P) where P * A = L * U

Solves the linear system A * x = b where A is an [n x n] matrix and b is a [n] column vector.

Solves the Continuous-time Lyapunov equation AP+PA\

Find the minimal polynomial of a simple algebraic expression. The minimal polynomial is the monic polynomial with rational coefficients of lowest degree for which the expression is a root. Supports: sqrt(n), cbrt(n), nthRoot(n, k), sqrt(a) + sqrt(b), and rational constants.

Compute the multivariate Taylor series expansion of a scalar expression around a point. Uses mixed partial derivatives to build the polynomial: f + sum_alpha (partial^|alpha| f / alpha!) * prod (xi - ai)^ki, where the sum is over all multi-indices of total degree <= order.

Convert an expression to canonical polynomial or rational normal form by expanding, combining fractions, cancelling common factors, and simplifying.

Solve first-order ordinary differential equations (ODEs) symbolically. The expression is the right-hand side of dy/dx = expr. Handles separable ODEs (dy/dx = g(x)*h(y)) and linear first-order ODEs (dy/dx + P*y = Q(x)) using integrating factors. Returns the general solution with constant C.

Compute higher-order and mixed partial derivatives of an expression. The variables argument specifies the differentiation order: a single string for first-order, or an array of strings for higher-order mixed partials.

Build and evaluate a piecewise-defined expression. Given an array of [condition, value] pairs and a default value, evaluates each condition in order and returns the value of the first matching condition, or the default if none match.

Add two polynomials element-wise. The shorter array is padded with zeros.

Compute the derivative of a polynomial given its coefficient array.

Multiply two polynomials by convolving their coefficient arrays.

Compute the Greatest Common Divisor (GCD) of two polynomials using the Euclidean algorithm. The result is normalized to be monic.

Compute the Least Common Multiple (LCM) of two polynomials. Computed as p*q/GCD(p,q). The result is normalized to be monic (leading coefficient 1).

Compute the quotient of polynomial long division p / q. Returns the quotient polynomial.

Compute the remainder of polynomial long division p / q. Returns the remainder polynomial.

Finds the roots of a univariate polynomial given by its coefficients starting from constant, linear, and so on, increasing in degree.

Evaluate a polynomial at a value x using Horner\

Expand powers and logarithms assuming arguments are positive real numbers. Distributes (a*b)^n->a^n*b^n, (a^m)^n->a^(m*n), log(a*b)->log(a)+log(b), and similar rules.

Calculates the Matrix QR decomposition. Matrix `A` is decomposed in two matrices (`Q`, `R`) where `Q` is an orthogonal matrix and `R` is an upper triangular matrix.

Transform a rationalizable expression in a rational fraction. If rational fraction is one variable polynomial then converts the numerator and denominator in canonical form, with decreasing exponents, returning the coefficients of numerator.

Solve an equation with domain constraints. Calls solve() and filters solutions by the specified domain or by assumptions stored via assume(). Valid domains: Positive, Negative, Nonnegative, Nonzero, Integer, Real, Complex, Rational.

Recursively substitute variables in an expression tree.

Compute the resultant of two univariate polynomials as the determinant of their Sylvester matrix. The resultant is zero if and only if the polynomials share a common root. Used for polynomial elimination.

Performs a real Schur decomposition of the real matrix A = UTU\

Compute the Taylor series expansion of an expression around a point. Uses the formula f(a) + f\

Extract the nth coefficient from the Taylor series expansion of an expression around a point. Uses the formula c_n = f^(n)(a) / n! where f^(n) is the nth derivative.

Simplify an expression tree.

Replace constant subexpressions of node with their values.

Perform simple one-pass simplifications on an expression tree.

/ unaryPlus: { trivial: T, total: T, commutative: T, associative: T }, /*

Calculate the Matrix LU decomposition with full pivoting. Matrix A is decomposed in two matrices (L, U) and two permutation vectors (pinv, q) where P * A * Q = L * U

Solve an equation or expression equal to zero for a given variable. Handles linear equations, quadratic equations (via quadratic formula), and simple polynomials (rational roots theorem). The input can be an equation like "x^2 - 4 = 0" or just an expression like "x^2 - 4" (treated as equal to zero).

Substitute a variable with a value or expression. Replaces all occurrences of the specified variable in an expression with the given value.

Compute the summation of an expression over a variable from start to end. Attempts closed-form evaluation for common patterns (arithmetic, quadratic, geometric series), then falls back to numeric summation.

Solves the real-valued Sylvester equation AX+XB=C for X

Returns true if the difference of the expressions simplifies to 0

Compute the product of an expression over a variable from start to end. Attempts closed-form evaluation for common patterns (factorial, partial factorial, constant product), then falls back to numeric evaluation.

Compute the tangent line of an expression at a given point. Returns a string expression y = f(a) + f\

Compute the Taylor polynomial expansion of an expression around a point. Returns only the polynomial string. Uses the formula f(a) + f\

Convert expressions containing fractional powers to explicit radical form. x^(1/2) becomes sqrt(x), x^(1/3) becomes cbrt(x), x^(p/q) becomes nthRoot(x^p, q). Numeric values are simplified where possible.

Combine fractions over a common denominator. This is the inverse of apart: it takes a sum of fractions and combines them into a single rational expression.

Expand trigonometric expressions using angle addition formulas: sin(a+b), cos(a+b), tan(a+b), and double-angle identities like sin(2*x).

Reduce products of trigonometric functions to sums using product-to-sum identities. Inverse of trigExpand. Applies sin^2, cos^2 half-angle and product formulas.

Convert trigonometric and hyperbolic functions to exponential form. Applies sin(x)->(exp(i*x)-exp(-i*x))/(2*i), cos(x)->(exp(i*x)+exp(-i*x))/2, and hyperbolic equivalents.

Finds one solution of the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector.

Finds all solutions of the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector.

Extract all symbolic variable names from an expression. Walks the AST and collects SymbolNode names, excluding known mathematical constants (pi, e, i) and built-in function names. Returns a sorted array of unique variable names.

Compute the Z-transform of a discrete sequence using a lookup table of known transform pairs. Supported: 1 → z/(z-1), n → z/(z-1)^2, n^2 → z*(z+1)/(z-1)^3, a^n → z/(z-a), n*a^n → a*z/(z-a)^2, cos(w*n) and sin(w*n) patterns.