- From basic operations,
1.1. evalf() outputs numerical value(s). [code lang="python"] >>> expr = sqrt(8) >>> expr 2 * sqrt(2) >>> expr.evalf() 2.82842712474619 [/code]
1.2. lambdify acts like lambda function. [code lang="python"] >>> import numpy >>> a = numpy.arange(10) >>> expr = sin(x) >>> f = lambdify(x, expr, "numpy") >>> f(a) [ 0. 0.84147098 0.90929743 0.14112001 -0.7568025 -0.95892427 -0.2794155 0.6569866 0.98935825 0.41211849] [/code]
1.3. lambbify for scalar input [code lang="python"] >>> f = lambdify(x, expr, "math") >>> f(0.1) 0.0998334166468 [/code]
- Simplify simplifies equations. Simplify runs slow. [code lang="python"] >>> simplify(sin(x)2 + cos(x)2) 1 >>> simplify*1 x - 1 >>> simplify(gamma(x)/gamma(x - 2)) (x - 2)⋅(x - 1) [/code]
2.1. simplification by expand() 式の展開 [code lang="python"] >>> expand*2 2 x - x - 6 [/code]
2.2. simplification by factor() 式の因数分解 [code lang="python"] >>> factor(x3 - x2 + x - 1) ⎛ 2 ⎞ (x - 1)⋅⎝x + 1⎠ >>> factor(x2z + 4xyz + 4*y2*z) 2 z⋅(x + 2⋅y) [/code]
2.3. simplification by collect() ある変数に関して乗数毎に項をまとめる [code lang="python"] >>> expr = xy + x - 3 + 2x2 - z*x2 + x**3 >>> expr 3 2 2 x - x ⋅z + 2⋅x + x⋅y + x - 3 >>> collected_expr = collect(expr, x) >>> collected_expr 3 2 x + x ⋅(-z + 2) + x⋅(y + 1) - 3 [/code]
2.4. simplification by cancel() 式を標準形(p/q，ここでp,qは共通項を持たない多項式)に変換． [code lang="python"] >>> cancel*3 x + 1 ───── x
>>> expr = 1/x + (3*x/2 - 2)/(x - 4) >>> expr 3⋅x ─── - 2 2 1 ─────── + ─ x - 4 x >>> cancel(expr) 2 3⋅x - 2⋅x - 8 ────────────── 2 2⋅x - 8⋅x
>>> expr = (xy**2 - 2xyz + xz2 + y2 - 2y*z + z2)/(x2 - 1) >>> expr 2 2 2 2 x⋅y - 2⋅x⋅y⋅z + x⋅z + y - 2⋅y⋅z + z ─────────────────────────────────────── 2 x - 1 >>> cancel(expr) 2 2 y - 2⋅y⋅z + z ─────────────── x - 1 [/code]
2.5. simplification by apart() 部分分数分解 [code lang="python"] >>> expr = (4x**3 + 21x2 + 10*x + 12)/(x4 + 5x**3 + 5x*2 + 4x) >>> expr 3 2 4⋅x + 21⋅x + 10⋅x + 12 ──────────────────────── 4 3 2 x + 5⋅x + 5⋅x + 4⋅x >>> apart(expr) 2⋅x - 1 1 3 ────────── - ───── + ─ 2 x + 4 x x + x + 1 [/code]
2.6. simplification by trigsimp() 三角関数の簡易化 [code lang="python"] >>> trigsimp(sin(x)2 + cos(x)2) 1 >>> trigsimp(sin(x)4 - 2*cos(x)2sin(x)2 + cos(x)4) cos(4⋅x) 1 ──────── + ─ 2 2 >>> trigsimp(sin(x)tan(x)/sec(x)) 2 sin (x)
>>> trigsimp(cosh(x)2 + sinh(x)2) cosh(2⋅x) >>> trigsimp(sinh(x)/tanh(x)) cosh(x) [/code]
2.7. simplification by powsimp() 乗数の整理 [code lang="python"] >>> powsimp(xa*xb) a + b x >>> powsimp(xa*ya) a (x⋅y)
>>> powsimp(tc*zc) c c t ⋅z >>> powsimp(tc*zc, force=True) c (t⋅z) [/code]
2.8. simplification by expand_power_() 乗数の分解 [code lang="python"] >>> expr = (xy)(a+b) a+b (x⋅y) >>> expand_power_exp(expr) a b (x⋅y) (x⋅y) >>> expand_power_base(expr) a+b (x⋅y) >>> expand_power_base(expr, force=True) a+b a+b x y [/code]
2.9. simplification by expand_log() log()の分解 [code lang="python"] >>> expand_log(log(xy)) log(x) + log(y) >>> expand_log(log(x/y)) log(x) - log(y) >>> expand_log(log(x2)) 2⋅log(x) >>> expand_log(log(xn)) n⋅log(x) >>> expand_log(log(zt)) log(t⋅z)
>>> expand_log(log(z2)) ⎛ 2⎞ log⎝z ⎠ >>> expand_log(log(z2), force=True) 2⋅log(z) [/code]
2.10. simplification by logcombine() log()の統合 [code lang="python"] >>> logcombine(log(x) + log(y)) log(x⋅y) >>> logcombine(nlog(x)) ⎛ n⎞ log⎝x ⎠ >>> logcombine(nlog(z)) n⋅log(z)
>>> logcombine(n*log(z), force=True) ⎛ n⎞ log⎝z ⎠ [/code]
