Pythonsf Calculations On Editors Supporting tagline

# PythonSf 式と その計算  
sin(1/`X+3`Y)(pi, pi/3)
===============================
-0.312961796208

sin(1/`X+3`Y)(`π, `π/3)
===============================
-0.312961796208

# Open 判ではプリプロセッサが貧弱なので積演算子の省略ができません。
sin(1/`X+3*`Y)(pi, pi/3)
===============================
-0.312961796208
sin(1/`X+3*`Y)(`π, `π/3)
===============================
-0.312961796208

# シンボリック微分: sin(1/x)
# PythonSf commercial 判
ts(); ∂x( ts.sin(1/`x) )
===============================
-cos(1/x)/x**2

# PythonSf open 判
ts(); x=ts.Symbol('x'); ts.diff( ts.sin(1/x), x)
===============================
-cos(1/x)/x**2

# Groebner 基底をシンボリックに計算する
# PythonSf commercial 判
ts();x,y,z=ts.symbols('x y z');f1,f2,f3=[x+y^2+3z^2-4, 4y^2-5,y^4-5z-10]; ts.groebner([f1,f2,f3], [x,y,z])
===============================
[-5/4 + y**2, 1483/256 + x, 27/16 + z]

# PythonSf open 判
ts();x,y,z=ts.symbols('x y z');f1,f2,f3=[x+y**2+3*z**2-4, 4*y**2-5,y**4-5*z-10]; ts.groebner([f1,f2,f3], [x,y,z])
===============================
[256*x + 1483, 4*y**2 - 5, 16*z + 27]

# 行列とベクトルの積演算
#  |1,2| * |5|   1/|1,2| * |5|
#  |3,4|   |6],    |3,4|   |6]
# PythonSf commercial 判
mt,vc=~[[1,2],[3,4]], ~[5,6]; mt vc, mt^-1 vc
===============================
(ClTensor([ 17.,  39.])
, ClTensor([-4. ,  4.5])

# PythonSf open 判
mt,vc=np.matrix([[1,2],[3,4]]), np.matrix([5,6]).T; mt*vc, np.linalg.inv(mt)*vc
===============================
(matrix([[17],
        [39]]), matrix([[-4. ],
        [ 4.5]]))

# x^2+1 の根が、四元数では無限個存在することの実験的証明
seed(0); vc=normalize(randn(3)); Oc(np.r_[0,vc])
===============================
Oc(0.0, 0.85771824099774296, 0.19456459875913087, 0.47588237619125601)

seed(0); vc=normalize(randn(3)); x=Oc(np.r_[0,vc]); x^2 + 1
===============================
Oc(1.11022302463e-16)

Z='1';S =λ s:s+'1';(λ s:λ z:s(s(z)))(S)(Z)
===============================
111

# Church 数での 2 + 3
Z='1';S =λ s:s+'1';(λ s:λ z:s(s(  s(s(s(z))) )))(S)(Z)
===============================
111111 Z:始めの数、任意の数に対する Successor を返す S だけから加算を実行する様子を具体的に示せています。 0→1, 1→11, 2→111, 3→1111, 4→11111, 5→111111 ... と対応させる一進方を使っています。

# 量子力学での調和振動子波動関数
P,X=:Px64,X64; H=P^2+10 X^2; vc=kzrs(64); vc[16]=1; renderMtCplx( ~[ expm(`i H t) vc for t in klsp(0, 5s`)]) 
# ８の字軌跡 inV=[-0.97m`,0.243m`, 0.97m`,-0.243m`, 0m`,0m`, -0.466m`/s`,-0.432m`/s`, -0.466m`/s`,-0.432m`/s`, 0.932m`/s`,0.864m`/s`]; N=len(inV)//4; getFV=λ v,i,k:(λ r=krry(v[2k:2k+2])-krry(v[2i:2i+2]):r/norm(r)^3 if norm(r)!=0 else ~[0,0])(); sumFc=λ v,j:sum([getFV(v,j,k) for k in range(N) if j!=k]); fnc= λ *v: np.r_[v[2N:],(~[sumFc(v,j) for j in range(N)]).r]; mt=kOde(fnc,inV, 2 s`,400); pt=plotTrajectory; pt(mt[:,:2]); pt(mt[:,2:4],color=red); pt(mt[:,4:6],color=green)
   
# ８の字軌跡で初期配置を z 軸方向に 0.1 だけずらした軌跡 D=3;inV=[-0.97m`,0.243,0, 0.97,-0.243,0, 0,0,0.1,  -0.466m`/s`,-0.432,0, -0.466,-0.432,0, 0.932,0.864,0]; N=len(inV)//(2D) ; getFV=λ v,i,k:(λ r=krry(v[D k:D (k+1)])-krry(v[D i:D (i+1)]):r/norm(r)^3 if norm(r)!=0 else ~[0,0])(); sumFc=λ v,j:sum([getFV(v,j,k) for k in range(N) if j!=k]); fnc= λ *v: np.r_[v[D N:],(~[sumFc(v,j) for j in range(N)]).r]; mt=kOde(fnc,inV, 20s`,400); cl=ClCplxColor(); for k in range(N):plotTrajectory(mt[:,D k:D (k+1)],color=cl.GetFltColor(exp(`i 2pi k/N)))

# ウラムの螺旋;;http://ja.wikipedia.org/wiki/ウラムの螺旋
N=10000; ts(); lsV=[~[1,0],~[0,1],~[-1,0],~[0,-1]]; Bf(last=~[0,0], len=1, nLen=0, n4=0, first=True); f=λ k:(Bf.last, Bf(last=Bf.last+lsV[Bf.n4], nLen=Bf.nLen+1), Bf(n4=Bf.n4) if Bf.nLen<Bf.len else [Bf(n4=(Bf.n4+1)%4, nLen=0), Bf(first=False) if Bf.first == True else Bf(first=True, len=Bf.len+1)])[0]; lsPt=tn.imap(f, tn.count())[:N]; mlb(); import pylab as pb; pb.scatter( *zip( *[pos for k,pos in enumerate(lsPt) if ts.isprime(k)])); pb.show();
↑1,3,5,7,9 奇数の線が出ている? やってみる

N=10000; ts(); lsV=[~[1,0],~[0,1],~[-1,0],~[0,-1]]; Bf(last=~[0,0], len=1, nLen=0, n4=0, first=True); f=λ k:(Bf.last, Bf(last=Bf.last+lsV[Bf.n4], nLen=Bf.nLen+1), Bf(n4=Bf.n4) if Bf.nLen<Bf.len else [Bf(n4=(Bf.n4+1)%4, nLen=0), Bf(first=False) if Bf.first == True else Bf(first=True, len=Bf.len+1)])[0]; lsPt=tn.imap(f, tn.count())[:N]; plotPt([pos for k,pos in enumerate(lsPt) if k%10 in [1,3,7,9] ])

np.matrix.__mro__
===============================
(<class 'numpy.matrixlib.defmatrix.matrix'>, <type 'numpy.ndarray'>, <type 'object'>)

dir(np.ndarray)
===============================
['T', '__abs__', '__add__', '__and__', '__array__', '__array_finalize__', '__array_interface__', '__array_prepare__', '__array_priority__', '__array_struct__', '__array_wrap__', '__class__', '__contains__', '__copy__', '__deepcopy__', '__delattr__', '__delitem__', '__delslice__', '__div__', '__divmod__', '__doc__', '__eq__', '__float__', '__floordiv__', '__format__', '__ge__', '__getattribute__', '__getitem__', '__getslice__', '__gt__', '__hash__', '__hex__', '__iadd__', '__iand__', '__idiv__', '__ifloordiv__', '__ilshift__', '__imod__', '__imul__', '__index__', '__init__', '__int__', '__invert__', '__ior__', '__ipow__', '__irshift__', '__isub__', '__iter__', '__itruediv__', '__ixor__', '__le__', '__len__', '__long__', '__lshift__', '__lt__', '__mod__', '__mul__', '__ne__', '__neg__', '__new__', '__nonzero__', '__oct__', '__or__', '__pos__', '__pow__', '__radd__', '__rand__', '__rdiv__', '__rdivmod__', '__reduce__', '__reduce_ex__', '__repr__', '__rfloordiv__', '__rlshift__', '__rmod__', '__rmul__', '__ror__', '__rpow__', '__rrshift__', '__rshift__', '__rsub__', '__rtruediv__', '__rxor__', '__setattr__', '__setitem__', '__setslice__', '__setstate__', '__sizeof__', '__str__', '__sub__', '__subclasshook__', '__truediv__', '__xor__', 'all', 'any', 'argmax', 'argmin', 'argsort', 'astype', 'base', 'byteswap', 'choose', 'clip', 'compress', 'conj', 'conjugate', 'copy', 'ctypes', 'cumprod', 'cumsum', 'data', 'diagonal', 'dot', 'dtype', 'dump', 'dumps', 'fill', 'flags', 'flat', 'flatten', 'getfield', 'imag', 'item', 'itemset', 'itemsize', 'max', 'mean', 'min', 'nbytes', 'ndim', 'newbyteorder', 'nonzero', 'prod', 'ptp', 'put', 'ravel', 'real', 'repeat', 'reshape', 'resize', 'round', 'searchsorted', 'setasflat', 'setfield', 'setflags', 'shape', 'size', 'sort', 'squeeze', 'std', 'strides', 'sum', 'swapaxes', 'take', 'tofile', 'tolist', 'tostring', 'trace', 'transpose', 'var', 'view']

info(fft)
info(np.linalg.inv)
 inv(a)

Compute the (multiplicative) inverse of a matrix.

Given a square matrix `a`, return the matrix `ainv` satisfying
``dot(a, ainv) = dot(ainv, a) = eye(a.shape[0])``.

Parameters
----------
a : array_like, shape (M, M)
    Matrix to be inverted.

Returns
-------
ainv : ndarray or matrix, shape (M, M)
    (Multiplicative) inverse of the matrix `a`.

Raises
------
LinAlgError
    If `a` is singular or not square.

Examples
--------
>>> from numpy import linalg as LA
>>> a = np.array([[1., 2.], [3., 4.]])
>>> ainv = LA.inv(a)
>>> np.allclose(np.dot(a, ainv), np.eye(2))
True
>>> np.allclose(np.dot(ainv, a), np.eye(2))
True

If a is a matrix object, then the return value is a matrix as well:

>>> ainv = LA.inv(np.matrix(a))
>>> ainv
matrix([[-2. ,  1. ],
        [ 1.5, -0.5]])
===============================
None ##■■ Python 環境としての PythonSf

# 数値の加減乗除べき乗算
(1-2*3/4)^5
===============================
-0.03125

# sin(1/x) value at x=pi
sin(1/`X)(pi)
===============================
0.312961796208

# 数値微分値: sin(1/x) value at x=pi
∂x( sin(2/`X) )(`π)
===============================
-0.162946719226

# シンボリック微分: sin(1/x)
ts(); ∂x( ts.sin(1/`x) )
===============================
-cos(1/x)/x**2

# Groebner 基底をシンボリックに計算する
ts();x,y,z=ts.symbols('x y z');f1,f2,f3=[x+y^2+3z^2-4, 4y^2-5,y^4-5z-10]; ts.groebner([f1,f2,f3], [x,y,z])
===============================
[-5/4 + y**2, 1483/256 + x, 27/16 + z]

# 行列とベクトルの積演算
#  |1,2| * |5|
#  |3,4|   |6]
mt,vc=~[[1,2],[3,4]], ~[5,6]; mt vc
===============================
[ 17.  39.]
---- ClTensor ----

# W:ワット、A:アンペア単位付きの数値計算
ts(); wattage,current=100W`,2.3A`; wattage/current^2
===============================
18.9035916824197*V`/A`

# Bool 体係数多項式の商と剰余
(`P^7+1)/(`P^2+1)
===============================
(Pl(`P^5+`P^3+`P), Pl(`P+1))

x=`P; (x^7+1)%(x^2+1)
===============================
`P+1

# 楕円関数:第一種の不完全/Jacobi 楕円関数の計算
φ,m=pi/3,0.6; sy(); u=ss.ellipkinc(φ,m); ss.ellipj(u,m)
===============================
(0.8660254037844386, 0.50000000000000011, 0.74161984870956643, 1.0471975511965976)

# Sn(4) 置換群における 置換集合:{Sb(2,3,0,1), Sb(1,0,3,2),Sb(1,3,2,0)} の左剰余類
=:SS4; SS4/{Sb(2,3,0,1), Sb(1,0,3,2),Sb(1,3,2,0)}
===============================
kfs([Sb(0,1,2,3), Sb(0,1,3,2), Sb(0,2,1,3), Sb(0,2,3,1), Sb(0,3,1,2), Sb(0,3,2,1)])

# 数値 1234 と文字列 '1' の hash コードを確認する Python テスト・コード
hash(1234), hash('1')
===============================
(1234, 1977051568)

# グラフ描画: sin(1/x) を [0.2, pi] 領域で
plotGr( sin(1/`X), 0.2, pi )
[1](#test name reference)

vim\gvim.exe

pi^0.5

===============================
1.77245385091

pi^0.5
===============================
1.77245385091

quadR(exp(-`X^2), -np.inf, np.inf)
===============================
1.77245385091

plotGr(λ t:quadR(exp(-`X^2), 0, t), 0,5)