SymPy学习之Numerical evaluation

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Basics

#N(expr, <args>)函数与sympify(expr).evalf(<args>)等价>>> from sympy import *>>> N(sqrt(2)*pi)4.44288293815837>>> (sqrt(2)*pi).evalf()4.44288293815837#额外的参数用于确认精确度>>> N(sqrt(2)*pi, 5)4.4429>>> N(sqrt(2)*pi, 50)4.4428829381583662470158809900606936986146216893757#支持复数>>> N(1/(pi + I), 20)0.28902548222223624241 - 0.091999668350375232456*I#有变量时进行部分计算>>> x = Symbol('x')>>> (pi*x**2 + x/3).evalf()3.14159265358979*x**2 + 0.333333333333333*x#也可以用标准python函数>>> float(pi)3.1415926535...>>> complex(pi+E*I)(3.1415926535...+2.7182818284...j)#理论上无精确度上限>>> str(N(pi, 10**6))[-50:] '95678796130331164628399634646042209010610577945815'

Floating-point numbers

>>> Float(0.1)0.100000000000000>>> Float(0.1, 10)0.1000000000>>> Float(0.125, 30)0.125000000000000000000000000000>>> Float(0.1, 30)   #传数值经精度比较低0.100000000000000005551115123126#解决方案>>> Float('0.1', 30)   #传字符串0.100000000000000000000000000000>>> Float(Rational(1, 10), 30)  #传Rational变量0.100000000000000000000000000000>>> Rational(1, 10).evalf(30)  #对Rational变量进行求值0.100000000000000000000000000000#以精确度较高的位数作为最终结果精确度>>> Float(0.1, 3)*Float(3.1415, 5)0.31417

Accuracy and error handling

#普通python计算无法进行高精度计算>>> a, b = GoldenRatio**1000/sqrt(5), fibonacci(1000)>>> float(a)4.34665576869e+208>>> float(b)4.34665576869e+208>>> float(a) - float(b)0.0#可以用N方法实现>>> N(fibonacci(100) - GoldenRatio**100/sqrt(5))-5.64613129282185e-22#精度更高时N方法也失效>>> N(fibonacci(1000) - (GoldenRatio)**1000/sqrt(5))0.e+85#加入maxn参数>>> N(fibonacci(1000) - (GoldenRatio)**1000/sqrt(5), maxn=500)-4.60123853010113e-210#高精度计算比较耗时，加入strict抛出错误>>> N(fibonacci(1000) - (GoldenRatio)**1000/sqrt(5), strict=True)#传入0时N函数无法得知，用chop参数剔除>>> f = fibonacci(100) - (GoldenRatio**100 - (GoldenRatio-1)**100)/sqrt(5)>>> N(f)0.e-104>>> N(f, maxn=1000)0.e-1336>>> N(f, chop=True)0>>> N(3 + I*f, chop=True)3.00000000000000#round函数四舍五入，保留指定位小数>>> v = 10*pi + cos(1)>>> N(v)31.9562288417661>>> v.round(3)31.956

Numerical simplification

#nsimplify用于化简，第二个参数为化简的类型，tolerance为误差>>> nsimplify(0.1)1/10>>> nsimplify(6.28, [pi], tolerance=0.01)2*pi>>> nsimplify(pi, tolerance=0.01)22/7>>> nsimplify(pi, tolerance=0.001)355---113>>> nsimplify(0.33333, tolerance=1e-4)1/3>>> nsimplify(2.0**(1/3.), tolerance=0.001)635---504>>> nsimplify(2.0**(1/3.), tolerance=0.001, full=True)3 ___\/ 2

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