Symbolic calculus: limits, symbolic matrices, first-order ODEs and the rational Risch algorithm

SymbolicLimit: the limit of an elementary expression, by substitution when the expression is continuous at the point, by the leading Taylor coefficients of numerator and denominator, by L'Hopital's rule for the indeterminate forms zero over zero and infinity over infinity (at most twelve differentiations), and at an infinite limit point by the exact degree comparison of a rational function, by the dominant monomial of an exp-log expression written as a sum of terms c exp(P(x)) x^a (log x)^b (Gruntz's most-rapidly-varying comparison restricted to that class), or by the squeeze theorem for a bounded numerator over an unbounded denominator. Anything else raises an error rather than returning a guess.

Usage

RischIntegration(expr, x = "x")

SymbolicLimit(expr, x = "x", x0 = 0, side = "both")

MatrixSymbolic(M, eigen_var = "t")

OdeSymbolic(ode = NULL, x = "x", y = "y", M = NULL, N = NULL)

Arguments

expr

Expression string.

x

Variable name (the integration or limit variable).

x0

Limit point; "inf", "-inf" and numbers are accepted.

side

"both", "+" or "-".

M

Square matrix of numbers or expression strings, or the M coefficient of a differential form in OdeSymbolic.

eigen_var

Name of the characteristic-polynomial variable.

ode

Right-hand side f(x, y), or "y' = f(x, y)".

y

Name of the dependent variable.

N

The N coefficient of a differential form.

Value

SymbolicLimit: list with limit, method, lhopital_steps and expression. MatrixSymbolic: list with determinant, inverse, char_poly, char_coeffs, eigenvalues, eigenvalues_numeric, trace and n. OdeSymbolic: list with classes, solution, form, coefficients, exponent and rhs. RischIntegration: list with integral, polynomial_part, rational_part, log_part, rational, out_of_scope, generators, verified and max_rel_error.

Details

MatrixSymbolic: determinant by Laplace cofactor expansion, inverse as the adjugate over the determinant, characteristic polynomial det(A - t I) expanded in eigen_var, and eigenvalues in closed form when that polynomial is linear or quadratic, numerically from the Durand-Kerner roots when all its coefficients are numbers, and not at all for a higher degree with symbolic entries (Abel-Ruffini).

OdeSymbolic: classification and closed-form solution of dy/dx = f(x, y) (or of a differential form M dx + N dy = 0): linear by the integrating factor exp(-int p dx), Bernoulli through v = y^(1-n), separable as the implicit int dy/h = int g dx + C, exact through the potential int M dx + int (N - d/dy int M dx) dy, and homogeneous reported with the substitution v = y/x. Each criterion is checked at five sample points.

RischIntegration: the rational case of the Risch-Bronstein algorithm: Euclidean division for the polynomial part, Hermite reduction (Yun squarefree decomposition and Bezout identities) for the rational part, and the Rothstein-Trager residues at the roots of the squarefree denominator for the logarithmic part, with conjugate pairs combined into real logarithms and arctangents. The transcendental tower and algebraic extensions are reported as out of scope rather than guessed.

The expression core, the shunting-yard parser and the integrator are copied from ExactAlgebra.R and SymbolicIntegration.R. Identical to the Python arm morie.fn.symcalc.

References

Risch, R. H. (1969). The problem of integration in finite terms. Transactions of the American Mathematical Society 139, 167-189.

Bronstein, M. (1997). Symbolic Integration I: Transcendental Functions. Springer, sections 2.2 and 2.4.

Yun, D. Y. Y. (1976). On square-free decomposition algorithms. Proceedings of SYMSAC 76, 26-35.

Gruntz, D. (1996). On Computing Limits in a Symbolic Manipulation System. PhD thesis, ETH Zurich.

Horn, R. A. and Johnson, C. R. (2013). Matrix Analysis, 2nd edn, sections 0.3 and 1.2. Cambridge University Press.

Boyce, W. E. and DiPrima, R. C. (2012). Elementary Differential Equations, 10th edn, sections 2.1, 2.2, 2.4 and 2.6. Wiley.

Examples

SymbolicLimit("sin(x)/x", "x", 0)$limit
#> [1] 1
SymbolicLimit("x*exp(-x)", "x", "inf")$limit
#> [1] 0
MatrixSymbolic(list(c("a", "b"), c("c", "d")))$determinant
#> [1] "a*d - b*c"
OdeSymbolic("dy/dx = 2*x*y")$solution
#> [1] "C*exp(x^2)"
RischIntegration("(x^2 + 1)/(x^3 - x)")$integral
#> [1] "log(x + 1) + log(x - 1) - log(x)"