Sympy ( Python )

Cos'è Sympy?

SymPy è una libreria di python dedicata al calcolo algebrico e all'analisi matematica. E' un modulo open source concesso con licenza BSD, quindi posso usarla liberamente e anche modificare il codice sorgente. Il che è un bel vantaggio.

Perché usare Sympy? A differenza di altri sistemi di risoluzione algebrica ( es. Maple, Mathematica ), Sympy non usa un proprio linguaggio sintattico bensì Python, il che riduce di parecchio i tempi di apprendimento. Basta conoscere Python per usarla.

Inoltre, essendo un modulo di Python, posso usare SymPy sia per il calcolo interattivo che nello sviluppo di procedure, applicazioni e programmi in ambiente python.

Come funziona Sympy
Come installare Sympy
Le funzioni

Come funziona Sympy

La libreria Sympy si basa sul calcolo simbolico.

Alcuni termini matematici sono trasformati in simboli ( es. variabili, radicali, ecc. ) in modo tale da non essere usati come numeri scalari.

Esempio

Nel calcolo numerico ogni radicale è sostituito dal suo scalare

$$ \sqrt{8} + \sqrt{8} = 2.828 + 2.828 \simeq 5.656 $$

Nel calcolo simbolico, invece, il radicale è considerato un simbolo.

Quindi, non viene calcolato il radicale.

$$ \sqrt{8} + \sqrt{8} = 2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2} $$

Il calcolo simbolico mi permette così di calcolare limiti, derivate, integrali, sistemi di equazioni, ecc.

Nota. Sympy è un componente interno usato da molti altri moduli come Cadabra, SageMath, PyDy, Glgebra, yt, StePy, Quameon, Lcapy, Quantum Programmin, LaTeX Expressions, Symbolic statistical modeling, OcSymPy, pyodesys, pyneqsys, ChemPy, Spyder.

Come installare Sympy

Per installare Sympy sull'interprete Python in ambiente Windows posso usare il package PIP o PIP3 nella directory Scripts di Python

pip install sympy

oppure su Linux

sudo pip install sympy

Il modulo Sympy è invece già incluso in Anaconda.

Le funzioni

Ecco un elenco parziale delle funzioni del modulo Sympy

Diff
Calcola la derivata di una funzione f(x)
Divisors
Trova i divisori di un numero intero
Limit
Calcola il limite di una funzione matematica f(x)
Solve
Trova le soluzioni di un'equazione

Altre funzioni

Abs
AccumBounds
Add
Adjoint
AlgebraicField
AlgebraicNumber
And
AppliedPredicate
Array
AssumptionsContext
Atom
AtomicExpr
BasePolynomialError
Basic
BlockDiagMatrix
BlockMatrix
CC
CRootOf
Catalan
Chi
Ci
Circle
CoercionFailed
Complement
ComplexField
ComplexRegion
ComplexRootOf
Complexes
ComputationFailed
ConditionSet
Contains
CosineTransform
Curve
DeferredVector
DenseNDimArray
Derivative
Determinant
DiagMatrix
DiagonalMatrix
DiagonalOf
Dict
DiracDelta
DisjointUnion
Domain
DomainError
DotProduct
Dummy
E
E1
EPath
EX
EXRAW
Ei
Eijk
Ellipse
EmptySequence
EmptySet
Eq
Equality
Equivalent
EulerGamma
EvaluationFailed
ExactQuotientFailed
Expr
ExpressionDomain
ExtraneousFactors
FF
FF_gmpy
FF_python
FU
FallingFactorial
FiniteField
FiniteSet
FlagError
Float
FourierTransform
FractionField
Function
FunctionClass
FunctionMatrix
GF
GMPYFiniteField
GMPYIntegerRing
GMPYRationalField
Ge
GeneratorsError
GeneratorsNeeded
GeometryError
GoldenRatio
GramSchmidt
GreaterThan
GroebnerBasis
Gt
HadamardPower
HadamardProduct
HankelTransform
Heaviside
HeuristicGCDFailed
HomomorphismFailed
I
ITE
Id
Identity
Idx
ImageSet
ImmutableDenseMatrix
ImmutableDenseNDimArray
ImmutableMatrix
ImmutableSparseMatrix
ImmutableSparseNDimArray
Implies
Indexed
IndexedBase
Integer
IntegerRing
Integers
Integral
Intersection
Interval
Inverse
InverseCosineTransform
InverseFourierTransform
InverseHankelTransform
InverseLaplaceTransform
InverseMellinTransform
InverseSineTransform
IsomorphismFailed
KroneckerDelta
KroneckerProduct
LC
LM
LT
Lambda
LambertW
LaplaceTransform
Le
LessThan
LeviCivita
Li
Limit
Line
Line2D
Line3D
Lt
MatAdd
MatMul
MatPow
Matrix
MatrixBase
MatrixExpr
MatrixPermute
MatrixSlice
MatrixSymbol
Max
MellinTransform
Min
Mod
Monomial
Mul
MultivariatePolynomialError
MutableDenseMatrix
MutableDenseNDimArray
MutableMatrix
MutableSparseMatrix
MutableSparseNDimArray
N
NDimArray
Nand
Naturals
Naturals0
Ne
NonSquareMatrixError
Nor
Not
NotAlgebraic
NotInvertible
NotReversible
Number
NumberSymbol
O
OmegaPower
OneMatrix
OperationNotSupported
OptionError
Options
Or
Order
Ordinal
POSform
Parabola
Permanent
PermutationMatrix
Piecewise
Plane
Point
Point2D
Point3D
PoleError
PolificationFailed
Poly
Polygon
PolynomialDivisionFailed
PolynomialError
PolynomialRing
Pow
PowerSet
PrecisionExhausted
Predicate
Product
ProductSet
PurePoly
PythonFiniteField
PythonIntegerRing
PythonRational
Q
QQ
QQ_I
QQ_gmpy
QQ_python
Quaternion
RR
Range
Rational
RationalField
Rationals
Ray
Ray2D
Ray3D
RealField
RealNumber
Reals
RefinementFailed
RegularPolygon
Rel
Rem
RisingFactorial
RootOf
RootSum
S
SOPform
SYMPY_DEBUG
Segment
Segment2D
Segment3D
SeqAdd
SeqFormula
SeqMul
SeqPer
Set
ShapeError
Shi
Si
Sieve
SineTransform
SingularityFunction
SparseMatrix
SparseNDimArray
StrPrinter
StrictGreaterThan
StrictLessThan
Subs
Sum
Symbol
SymmetricDifference
SympifyError
TableForm
Trace
Transpose
Triangle
TribonacciConstant
Tuple
Unequality
UnevaluatedExpr
UnificationFailed
Union
UnivariatePolynomialError
UniversalSet
Wild
WildFunction
Xor
Ynm
Ynm_c
ZZ
ZZ_I
ZZ_gmpy
ZZ_python
ZeroMatrix
Znm
__all__
__builtins__
__cached__
__doc__
__file__
__loader__
__name__
__package__
__path__
__spec__
__sympy_debug
__version__
abundance
acos
acosh
acot
acoth
acsc
acsch
adjoint
airyai
airyaiprime
airybi
airybiprime
algebras
apart
apart_list
appellf1
apply_finite_diff
approximants
are_similar
arg
arity
asec
asech
asin
asinh
ask
assemble_partfrac_list
assoc_laguerre
assoc_legendre
assuming
assumptions
atan
atan2
atanh
banded
bell
bernoulli
besseli
besselj
besselk
besselsimp
bessely
beta
betainc
betainc_regularized
binomial
binomial_coefficients
binomial_coefficients_list
block_collapse
blockcut
bool_map
bottom_up
bspline_basis
bspline_basis_set
cacheit
calculus
cancel
capture
carmichael
cartes
casoratian
catalan
cbrt
ccode
ceiling
centroid
chebyshevt
chebyshevt_poly
chebyshevt_root
chebyshevu
chebyshevu_poly
chebyshevu_root
check_assumptions
checkodesol
checkpdesol
checksol
classify_ode
classify_pde
closest_points
cofactors
collect
collect_const
combsimp
comp
compose
composite
compositepi
concrete
conjugate
construct_domain
content
continued_fraction
continued_fraction_convergents
continued_fraction_iterator
continued_fraction_periodic
continued_fraction_reduce
convex_hull
convolution
core
cos
cosh
cosine_transform
cot
coth
count_ops
count_roots
covering_product
csc
csch
cse
cxxcode
cycle_length
cyclotomic_poly
decompogen
decompose
default_sort_key
deg
degree
degree_list
denom
derive_by_array
det
det_quick
diag
diagonalize_vector
dict_merge
diff
difference_delta
differentiate_finite
digamma
diophantine
dirichlet_eta
discrete
discrete_log
discriminant
div
divisor_count
divisor_sigma
doctest
dotprint
dsolve
egyptian_fraction
elliptic_e
elliptic_f
elliptic_k
elliptic_pi
epath
erf
erf2
erf2inv
erfc
erfcinv
erfi
erfinv
euler
euler_equations
evalf
evaluate
exp
exp_polar
expand
expand_complex
expand_func
expand_log
expand_mul
expand_multinomial
expand_power_base
expand_power_exp
expand_trig
expint
exptrigsimp
exquo
external
eye
factor
factor_list
factor_nc
factor_terms
factorial
factorial2
factorint
factorrat
failing_assumptions
false
farthest_points
fcode
ff
fft
fibonacci
field
field_isomorphism
filldedent
finite_diff_weights
flatten
floor
fourier_series
fourier_transform
fps
frac
fraction
fresnelc
fresnels
fu
functions
fwht
gamma
gammasimp
gcd
gcd_list
gcd_terms
gcdex
gegenbauer
genocchi
geometry
get_contraction_structure
get_indices
gff
gff_list
glsl_code
grevlex
grlex
groebner
ground_roots
group
gruntz
hadamard_product
half_gcdex
hankel1
hankel2
hankel_transform
harmonic
has_dups
has_variety
hermite
hermite_poly
hessian
hn1
hn2
homogeneous_order
horner
hyper
hyperexpand
hypersimilar
hypersimp
idiff
ifft
ifwht
igcd
igrevlex
igrlex
ilcm
ilex
im
imageset
init_printing
init_session
integer_log
integer_nthroot
integrals
integrate
interactive
interactive_traversal
interpolate
interpolating_poly
interpolating_spline
intersecting_product
intersection
intervals
intt
inv_quick
inverse_cosine_transform
inverse_fourier_transform
inverse_hankel_transform
inverse_laplace_transform
inverse_mellin_transform
inverse_mobius_transform
inverse_sine_transform
invert
is_abundant
is_amicable
is_convex
is_decreasing
is_deficient
is_increasing
is_mersenne_prime
is_monotonic
is_nthpow_residue
is_perfect
is_primitive_root
is_quad_residue
is_strictly_decreasing
is_strictly_increasing
is_zero_dimensional
isolate
isprime
itermonomials
jacobi
jacobi_normalized
jacobi_poly
jacobi_symbol
jn
jn_zeros
jordan_cell
jscode
julia_code
kronecker_product
kroneckersimp
laguerre
laguerre_poly
lambdify
laplace_transform
latex
lcm
lcm_list
legendre
legendre_poly
legendre_symbol
lerchphi
lex
li
limit
limit_seq
line_integrate
linear_eq_to_matrix
linsolve
list2numpy
ln
log
logcombine
loggamma
logic
lowergamma
lucas
maple_code
marcumq
mathematica_code
mathieuc
mathieucprime
mathieus
mathieusprime
mathml
matrices
matrix2numpy
matrix_multiply_elementwise
matrix_symbols
maximum
meijerg
mellin_transform
memoize_property
mersenne_prime_exponent
minimal_polynomial
minimum
minpoly
mobius
mobius_transform
mod_inverse
monic
motzkin
multigamma
multiline_latex
multinomial_coefficients
multipledispatch
multiplicity
n_order
nan
nextprime
nfloat
nonlinsolve
not_empty_in
npartitions
nroots
nsimplify
nsolve
nth_power_roots_poly
ntheory
nthroot_mod
ntt
numbered_symbols
numer
octave_code
ode_order
ones
oo
ord0
ordered
pager_print
parallel_poly_from_expr
parse_expr
parsing
partition
pde_separate
pde_separate_add
pde_separate_mul
pdiv
pdsolve
per
perfect_power
periodic_argument
periodicity
permutedims
pexquo
pi
piecewise_fold
plot
plot_backends
plot_implicit
plot_parametric
plotting
polar_lift
polarify
pollard_pm1
pollard_rho
poly
poly_from_expr
polygamma
polylog
polys
posify
postfixes
postorder_traversal
powdenest
powsimp
pprint
pprint_try_use_unicode
pprint_use_unicode
pquo
prefixes
prem
preorder_traversal
pretty
pretty_print
preview
prevprime
prime
prime_decomp
prime_valuation
primefactors
primenu
primeomega
primepi
primerange
primitive
primitive_element
primitive_root
primorial
principal_branch
print_ccode
print_fcode
print_glsl
print_gtk
print_jscode
print_latex
print_maple_code
print_mathml
print_python
print_rcode
print_tree
printing
prod
product
proper_divisor_count
proper_divisors
public
pycode
python
quadratic_congruence
quadratic_residues
quo
rad
radsimp
randMatrix
random_poly
randprime
rational_interpolate
ratsimp
ratsimpmodprime
rcode
rcollect
re
real_root
real_roots
reduce_abs_inequalities
reduce_abs_inequality
reduce_inequalities
reduced
reduced_totient
refine
refine_root
register_handler
release
rem
remove_handler
reshape
residue
resultant
rf
riemann_xi
ring
root
rootof
roots
rot_axis1
rot_axis2
rot_axis3
rotations
round_two
rsolve
rsolve_hyper
rsolve_poly
rsolve_ratio
rust_code
satisfiable
sec
sech
separatevars
sequence
series
seterr
sets
sfield
shape
sieve
sift
sign
signsimp
simplify
simplify_logic
sin
sinc
sine_transform
singularities
singularityintegrate
sinh
solve_linear
solve_linear_system
solve_linear_system_LU
solve_poly_inequality
solve_poly_system
solve_rational_inequalities
solve_triangulated
solve_undetermined_coeffs
solve_univariate_inequality
solvers
solveset
source
sqf
sqf_list
sqf_norm
sqf_part
sqrt
sqrt_mod
sqrt_mod_iter
sqrtdenest
srepr
sring
sstr
sstrrepr
stationary_points
stieltjes
strategies
sturm
subfactorial
subresultants
subsets
substitution
summation
swinnerton_dyer_poly
symarray
symbols
symmetric_poly
symmetrize
sympify
take
tan
tanh
tensor
tensorcontraction
tensordiagonal
tensorproduct
terms_gcd
test
testing
textplot
threaded
timed
to_cnf
to_dnf
to_nnf
to_number_field
together
topological_sort
total_degree
totient
trace
trailing
transpose
tribonacci
trigamma
trigsimp
true
trunc
unbranched_argument
unflatten
unpolarify
uppergamma
use
utilities
var
variations
vectorize
vfield
viete
vring
wronskian
xfield
xring
xthreaded
yn
zeros
zeta
zoo