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