Maple 2019 includes numerous cutting-edge updates in a variety of branches of mathematics.
Real Roots of Polynomials
The new command FindODE, in the DEtools package, tries to find a linear ordinary differential equation with polynomial coefficients for the given expression.
DEtoolsFindODEBesselI0,x + x BesselI2,x, yx
There have been various improvements made to the int command for Maple 2019.
New results from int:
Improved answers for definite integrals when the AllSolutions option is given:
introundt,t=0..x,AllSolutions # Similar improvements with ceil, floor, frac, trunc instead of round.
The inttrans package in Maple 2019 has had several transforms, specifically laplace, invlaplace, fourier and invfourier, extended to handle a larger class of problems, and in some cases already handled classes of problems faster. This has been accomplished via an integration by differentiation approach described in the following:
- A. Kempf, D.M. Jackson and A.H. Morales, "New Dirac delta function based methods with applications to perturbative expansions in quantum field theory", J. Phys. A:47, 2014
- D. Jia, E. Tang, and A. Kempf, "Integration by differentiation: new proofs, methods and examples", J. Phys. A:50, 2017
One can view this approach, in simplest possible terms, as a product rule.
Here are a few examples which failed to transform in prior versions of Maple, but now transform quite rapidly:
The limit command in Maple 2019 has been enhanced for the case of limits of quotients of multivariate functions. See Multivariate Limits for details.
A new algorithm for univariate polynomials has been added to the RootFinding:-Isolate command. It is particularly efficient for ill-conditioned problems and high accuracy solutions, and it provides certified real root isolation for polynomials with irrational coefficients. See Real Root Finding for details.
The residue command has a new optional argument that allows the user to specify the maximal order of the underlying series computations. See residue for an example.
The simplify command in Maple 2019 has undergone several improvements, especially with regard to expressions containing piecewise functions.
Simplification of expressions containing piecewise functions has been improved.
Equal, equivalent, or implied piecewise branches are now combined by simplify;
simplifym2m < 20otherwiseassuming m∷posint
Piecewise conditions involving floor, ceil, round, frac, trunc can now be simplified:
Branch conditions other than equations, inequations, and inequalities are now taken into account while simplifying branch values:
Branch conditions are now simplified more effectively using basic boolean logic:
simplify now reorders piecewise conditions when appropriate:
Piecewise conditions are now better normalized;
Common terms and factors are now pulled out of piecewise branch values where possible:
Nonpiecewise-related improvements made to simplify:
Improved simplification of Gamma:
simplify1Γ⁡1−n⁢Γ⁡n assuming n∷integer
Trig functions are now expanded if it helps with simplification:
Simplification of expressions containing arctan has been improved:
Expressions containing csgn can now be more effectively simplified:
Conversion between powers, exponentials, trig functions, and radicals to achieve simplification has been improved:
Symbolic powers of integers are now combined more effectively:
simplify now rewrites expressions using a common integer base:
Radicals are now typically combined by simplify:
If appropriate conditions are satisfied, certain simplifications of floor, ceil, and round are applied:
EllipticE−1n,x now simplifies:
The solve command in Maple 2019 has undergone several improvements.
Maple2019 solves equations with inequalities more carefully:
There are other commands which have improved.
minimize can now solve this example:
expand now takes into account more assumptions:
floor and ceil now make better use of assumptions:
rationalize works better on certain examples of nested radicals:
Expressions with nested calls to Re and Im now evaluate better:
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