# SymPy 0.7.4¶

9 Dec 2013

## Major changes¶

Python 3

SymPy now uses a single code-base for Python 2 and Python 3.

Geometric Algebra

The internal representation of a multivector has been changes to more fully use the inherent capabilities of SymPy. A multivector is now represented by a linear combination of real commutative SymPy expressions and a collection of non-commutative SymPy symbols. Each non-commutative symbol represents a base in the geometric algebra of an N-dimensional vector space. The total number of non-commutative bases is

`2**N - 1`

(`N`

of which are a basis for the vector space) which when including scalars give a dimension for the geometric algebra of`2**N`

. The different products of geometric algebra are implemented as functions that take pairs of bases symbols and return a multivector for each pair of bases.The LaTeX printing module for multivectors has been rewritten to simply extend the existing sympy LaTeX printing module and the sympy LaTeX module is now used to print the bases coefficients in the multivector representation instead of writing an entire LaTeX printing module from scratch.

The main change in the geometric algebra module from the viewpoint of the user is the interface for the gradient operator and the implementation of vector manifolds:

The gradient operator is now implemented as a special vector (the user can name it

`grad`

if they wish) so the if`F`

is a multivector field all the operations of`grad`

on`F`

can be written`grad*F`

,`F*grad`

,`grad^F`

,`F^grad`

,`grad|F`

,`F|grad`

,`grad<F`

,`F<grad`

,`grad>F`

, and`F>grad`

where`**`

,`^`

,`|`

,`<`

, and`>`

are the geometric product, outer product, inner product, left contraction, and right contraction, respectively.The vector manifold is defined as a parametric vector field in an embedding vector space. For example a surface in a 3-dimensional space would be a vector field as a function of two parameters. Then multivector fields can be defined on the manifold. The operations available to be performed on these fields are directional derivative, gradient, and projection. The weak point of the current manifold representation is that all fields on the manifold are represented in terms of the bases of the embedding vector space.

Classical Cryptography, implements:

Affine ciphers

Vigenere ciphers

Bifid ciphers

Hill ciphers

RSA and “kid RSA”

linear feedback shift registers.

Common Subexpression Elimination (CSE). Major changes have been done in cse internals resulting in a big speedup for larger expressions. Some changes reflect on the user side:

Adds and Muls are now recursively matched (

`[w*x, w*x*y, w*x*y*z]`

ǹow turns into`[(x0, w*x), (x1, x0*y)], [x0, x1, x1*z]`

)CSE is now not performed on the non-commutative parts of multiplications (it avoids some bugs).

Pre and post optimizations are not performed by default anymore. The

`optimizations`

parameter still exists and`optimizations='basic'`

can be used to apply previous default optimizations. These optimizations could really slow down cse on larger expressions and are no guarantee of better results.An

`order`

parameter has been introduced to control whether Adds and Muls terms are ordered independently of hashing implementation. The default`order='canonical'`

will independently order the terms.`order='none'`

will not do any ordering (hashes order is used) and will represent a major performance improvement for really huge expressions.In general, the output of cse will be slightly different from the previous implementation.

Diophantine Equation Module. This is a new addition to SymPy as a result of a GSoC project. With the current release, following five types of equations are supported.

Linear Diophantine equation, \(a_{1}x_{1} + a_{2}x_{2} + . . . + a_{n}x_{n} = b\)

General binary quadratic equation, \(ax^2 + bxy + cy^2 + dx + ey + f = 0\)

Homogeneous ternary quadratic equation, \(ax^2 + by^2 + cz^2 + dxy + eyz + fzx = 0\)

Extended Pythagorean equation, \(a_{1}x_{1}^2 + a_{2}x_{2}^2 + . . . + a_{n}x_{n}^2 = a_{n+1}x_{n+1}^2\)

General sum of squares, \(x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 = k\)

Unification of Sum, Product, and Integral classes

A new superclass has been introduced to unify the treatments of indexed expressions, such as Sum, Product, and Integral. This enforced common behavior accross the objects, and provides more robust support for a number of operations. For example, Sums and Integrals can now be factored or expanded.

`S.subs()`

can be used to substitute for expressions inside a Sum/Integral/Product that are independent of the index variables, including unknown functions, for instance,`Integral(f(x), (x, 1, 3)).subs(f(x), x**2)`

, while`Sum.change_index()`

or`Integral.transform`

are now used for other changes of summation or integration variables. Support for finite and infinite sequence products has also been restored.In addition there were a number of fixes to the evaluation of nested sums and sums involving Kronecker delta functions, see sympy/sympy#7023 and sympy/sympy#7086.

Series

The

`Order`

object used to represent the growth of a function in series expansions as a variable tend to zero can now also represent growth as a variable tend to infinity. This also fixed a number of issues with limits. See sympy/sympy#3333 and sympy/sympy#5769.Division by

`Order`

is disallowed, see sympy/sympy#4855.Addition of

`Order`

object is now commutative, see sympy/sympy#4279.

Physics

Initial work on gamma matrices, depending on the tensor module.

Logic

New objects

`true`

and`false`

which are`Basic`

versions of the Python builtins`True`

and`False`

.

Other

Arbitrary comparisons between expressions (like

`x < y`

) no longer have a boolean truth value. This means code like`if x < y`

or`sorted(exprs)`

will raise`TypeError`

if`x < y`

is symbolic. A typical fix of the former is`if (x < y) is True`

(assuming the`if`

block should be skipped if`x < y`

is symbolic), and of the latter is`sorted(exprs, key=default_sort_key)`

, which will order the expressions in an arbitrary, but consistent way, even across platforms and Python versions. See sympy/sympy#5931.Arbitrary comparisons between complex

*numbers*(for example,`I > 1`

) now raise`TypeError`

as well (see sympy/sympy#2510).`minimal_polynomial`

now works with algebraic functions, like`minimal_polynomial(sqrt(x) + sqrt(x + 1), y)`

.`exp`

can now act on any matrix, even those which are not diagonalizable. It is also more comfortable to call it,`exp(m)`

instead of just`m.exp()`

, as was required previously.`sympify`

now has an option`evaluate=False`

that will not automatically simplify expressions like`x+x`

.Deep processing of

`cancel`

and`simplify`

functions.`simplify`

is now recursive through the expression tree. See e.g. sympy/sympy#7022.Improved the modularity of the codebase for potential subclasses, see sympy/sympy#6751.

The SymPy cheatsheet was cleaned up.

## Compatibility breaks¶

Removed deprecated Real class and is_Real property of Basic, see sympy/sympy#4820.

Removed deprecated ‘each_char’ option for

`symbols()`

, see sympy/sympy#5018.The

`viewer="StringIO"`

option to`preview()`

has been deprecated. Use`viewer="BytesIO"`

instead. See sympy/sympy#7083.`TransformationSet`

has been renamed to`ImageSet`

. Added public facing`imageset`

function.