Maple

• The simplify command has undergone several improvements, in particular with regard to expressions containing piecewise functions.
• Improvements to integration mean Maple 2019 can solve more integrals than previously.
• The solve command in Maple 2019 has been improved, including enhancements when solving inequalities.
• A new command, FindODE, in the DEtools package, tries to find a linear ordinary differential equation with polynomial coefficients for the given expression
• The commands for performing Laplace, inverse Laplace, Fourier, and inverse Fourier transforms can handle a larger class of problems, and some classes of problems are handled faster than before.
• A new algorithm for univariate polynomials has been added to the RootFinding:-Isolate command, which is particularly efficient for ill-conditioned problems and for finding high-accuracy solutions, and provides certified real root isolation for polynomials with irrational coefficients.
• The residue command has a new optional argument that allows the user to specify the maximal order of the underlying series computations.
• Rationalize works better on certain examples of nested radicals.
• Expressions with nested calls to Re and Im now give improved results.
• Expand, floor, and ceiling have more support for assumptions.

Maple 2019 includes significant improvements that further strengths Maple's world-leading ability in finding exact solutions to partial differential equations, including additional methods for solving new classes of problems, more flexibility in choosing the methods to try, and improved simplification of the results. Improvements have been made in the following areas:

• PDE and boundary condition problems solved using linear change of variables
• Specifying or excluding methods for solving
• Series solutions for linear PDE and BC problems solved via product separation with eigenvalues that are the roots of algebraic expressions which cannot be inverted
• Superposition method for linear PDE with more than one non-homogeneous BC
• Polynomial solutions method
• Solving more problems using the Laplace transform or the Fourier transform
• Improvements to solving heat and wave PDEs, with or without a source
• Improvements in series methods for Laplace PDE problems
• Better simplification of answers
• Linear differential operator: more solutions are now successfully computed
• More problems in three variables are now solved

The Root Finding package is a suite of advanced commands for finding roots numerically. The routines in this package augment the root-finding functionality provided by fsolve, especially with respect to finding several roots concurrently, so all improvements to the Root Finding package automatically result in improvements to fsolve.



The RootFinding:-Isolate command has been expanded so that it can isolate the roots of univariate polynomials with arbitrary real coefficients, not just rationals or floats. In particular, Isolate can now be used to find roots of polynomials with algebraic coefficients.

The new default algorithm of Isolate also features vastly improved performance for ill-conditioned polynomials with clustered roots. The root finding method eventually converges quadratically to regions containing roots, rather than just linearly. The same technique allows even more dramatic improvements for root finding requests with high accuracy even on well-conditioned problems.

Maple provides a state-of-the-art environment for algebraic computations in physics, with emphasis on ensuring that the computational experience is as natural as possible. Maple 2019 both consolidates the functionality introduced in previous releases, and provides significant enhancements that further strengthen its functionality, especially in quantum mechanics, tensor computations, and documentation.

Highlights from quantum mechanics include coherent states, tensor products of states, Taylor series of expressions involving anti-commutative variables and functions, and several improvements involving the normalization and simplification of expressions taking into account commutator and anti-commutator algebra rules.

Significant work was done in Maple 2019 to further strengthen, consolidate, and polish Maple’s support for tensor computations, making Maple unmatched in the field, covering classical and quantum mechanics, and special and general relativity. In addition, Maple supports natural tensor notation on both input and output, tight integration with the full Maple computation system, and extensive supporting documentation. In particular, A Complete Guide for Performing Tensor Computations using Physics is a new e-book included in Maple 2019. This guide covers tensors and their use in Euclidean spaces, special relativity, quantum mechanics, and classical field theory, as well as general relativity and transformations of coordinates on tensorial expressions.

In addition to the new tensor guide, Maple 2019 also includes a mini-course, Computer Algebra for Physicists, as well as a new Physics Updates page that consolidates and augments examples, presentations, relevant blog posts, and information from previous updates into a more convenient single page.

• The new FindDessins command computes all dessins d'enfants corresponding to a specified branching pattern, and the DecomposeDessin command finds all decompositions of the corresponding Belyi map.
• Maple is now able to recognize finite Frobenius groups, either as abstract groups, or as permutation groups, and to compute Frobenius kernels and complements.
• Several new commands allow you to work with finite Hamiltonian groups.
• New commands enable you to compute various subgroup series and study their properties, including composition series, ordered Sylow towers, and Frattini series.
• The new DirectFactors command produces a Remak decomposition of a finite group.
• You can now test whether a finite group has various properties, such as Hamiltonian, Frobenius, directly indecomposable, metacyclic, or if a finite group belongs to various other classes of soluable groups.
• The SmallGroup constructor can now construct groups of order p^k for arbitrary primes p and positive integers k <= 4.
• New number-theoretic commands allow you to determine whether every group of a given order has a specific property, such as cyclic, abelian, soluable, and more.
• Maple 2019 includes a new database of Frobenius groups, along with commands for constructing, identifying, and manipulating them.
• You can now search the database of perfect groups for groups satisfying various combinations of properties, including the order, center, socle, Fitting subgroup, class number and others.
• Many new properties have been added to the databases of small groups and transitive groups.
• Operations on permutation groups are significantly faster and use much less memory.

Computational geometry problems occur in many applications involving points in two- or higher-dimensional spaces, such as feature recognition, predicting vapor-liquid phase diagrams, delineating closely related regions for scattered data, and more. Maple’s Computational Geometry package, which applies computational methods to polygons and clouds of points, has been expanded in Maple 2019, with new tests of properties and computations abilities.

In particular, Maple 2019 includes efficient tests for determining point orientation, if a point lies in a circle or on a line segment defined by other points, if two line segments intersect, or if collections of line segments have intersections. You can also find the closest pair of points in a collection, and the DelaunayTriangulation command now supports computing triangularizations in dimensions up to 10.

For many commands, you have the option of displaying a graphical representation of the steps of the algorithm.

Maple 2019 offers a variety of improvements for handing, analyzing, and visualizing your data.

• Maple 2019 can now compute least trimmed squares regressions.
• The Correlogram command computes autocorrelations of a data set and displays the result as a column plot with dashed lines indicating the lower and upper 95% confidence bands.
• The Detrend command removes any trend from a set of data.
• The Difference command computes lagged differences between elements in a data set.
• The display of data from large objects, including matrices, data frames, and data series, has been improved so that you can easily view the first few rows and columns of the data.
• Several commands have been updated to support DataFrames and DataSeries, including remove, select and selectremove.
• The Biplot command has a new option that enables you to select the principal components used in the biplot.
• The DataSummary, FivePointSummary, and FrequencyTable commands have a new option to control the relative column widths in the resulting table.

• Generate periodograms of data sampled at irregular time intervals.
• Find the peaks and valleys of 1-D data sets, including filtering out peaks or valleys that are too close, specifying what defines a peak or valley, and more.
• Specify the overlap when generating a spectrogram.
• Calculate the real cepstrum, complex cepstrum and inverse complex cepstrum of a signal.
• Swap data in a matrix or a vector into a different position using the new FFTShift command, such as moving the lowest frequencies to the center and the highest frequencies to the corners to produce more meaningful visualizations and simplified manipulations of data.
• Apply an edge detection convolution mask to an image, including Sobel, Robert, Prewitt 3x3, and Prewitt 4x4.
• Calculate the magnitude of the Fourier transform of an image, which can then be embedded as an image to visualize the spatial frequencies of the image.

A completely redesigned Start Page, as well as an expanded set of user resources, provides a more helpful environment for everyone when you open Maple

• The Start Page includes new and improved resources designed to help new users get started quickly, including a short introductory video.
• For more experienced users, the Start Page has been simplified to focus on the most helpful, frequently used resources.
• The Maple Portal, a comprehensive resource for any user, has been expanded to present both in-product and web-based resources, organized by preferred learning style.
• For those who would prefer to start from a blank document, it is now significantly easier to turn the Start Page off.

Maple contains many tools designed explicitly to help students learn new ideas, explore concepts, and solve problems, all in an easy-to-use environment. Maple 2019 includes several additions to student-specific resources, including:

• Many more operations from Numerical Analysis can now be performed at the click of a button using menus in the context panel, such as computing the rate of convergence of a sequence or function, finding Taylor polynomial approximations, computing the limit of a vector, calculating the relative error, and more.
• A new assistant makes it easier to create randomly generated and automatically graded practice sheets, for drilling problems in arithmetic, algebra, calculus, factorization and more.
• New interactive Math Apps help students understand important concepts from computer science, such as sorting algorithms and big O notation, as well as topics from other fields.

Maple 2019 includes a variety of improvements to the interface to improve your workflow.

• Maple 2019 makes it much easier to get a sense of the content of large matrices, vectors, arrays, data frames, and data series, by displaying the first 10 rows and 10 columns of the data, as well as the size of the structure.
• The new Units by Dimensionality palette expands the number of available unit templates and makes it easier to find units associated with physical dimensions.
• The entries in the Favorites palette can be easily rearranged by clicking and dragging, and in addition, you can now create custom entries.
• New options to the interface command give more control over the display of procedures, colors of output when using the command-line interface, and more.
• Various menus and toolbars have been simplified.
• More operations from the Student Numerical Analysis package can be performed using the context panel.

The Maple Quantum Chemistry Toolbox from RDMChem, a new add-on toolbox for Maple 2019, provides a comprehensive, easy-to-use environment for the parallel computation of the electronic energies and properties of molecules. With this toolbox, you can:

• Define molecules instantly from a database of more than 96 million molecules
• Run quantum computations with well-known electronic structure methods as well as recently developed, advanced methods for cutting-edge research
• Analyze molecular energies and properties through publication-quality, 2-D and 3-D plots and animations.