k. cooper - washington state university

Limits Differentiation Integration Solving ODEs

Machine Calculus

K. Cooper1

1Department of MathematicsWashington State University

2018

Cooper Washington State University

Symbolic Calculus


Sympy Calculus

Sympy has a full array of Integral and Differential Calculuscapability.

It can find limits, derivatives, antiderivatives, evaluate Taylorseries, and solve differential equations.

It can solve linear and nonlinear systems of equations.


Symbolic Calculus


Limits

Sympy loves to compute limits.

sp.limit(sp.sin(x)/x,x,0)produces an answer of 1.

sp.limit((3*x*x+2*x-1)/(2*x*x-101*x),x,sp.oo)gives 3/2.

Note that infinity in Sympy is called oo. One can alsosometimes use an IEEE 754 infinity, called inf.


Symbolic Calculus


Limits

We can also do one-sided limits:

sp.limit(1/x,x,0,’+’)oosp.limit(1/x,x,0,’-’)-oo


Symbolic Calculus


Differentiation

The basic command is diff.diff(expr,var)

You can leave off the variable if it is clear.diff(x*x+2*x+1)gives 2*x + 2, butdiff(alpha*x*x+2*x+1)gives an error:Since there is more than one variable in the expression, the

variable(s) of differentiation must be supplied to differentiate

diff(alpha*x*x+2*x+1,x)produces 2*alpha*x + 2


Symbolic Calculus


Derivatives

We could call the diff function as a method of the expression.

(alpha*x*x+2*x+1).diff(x)yields the obvious answer.

To get higher order derivatives, we just add one moreargument: the order of the derivative.

diff(alpha*x*x+2*x+1,x,2)(alpha*x*x+2*x+1).diff(x,2)both give 2*alpha.


Symbolic Calculus


Example: Newton’s Method

Find all points where sin(x) has value 34 .

def ntn_mth(expr,var,x0,tol):deriv = expr.diff(var)digits = -log10(tol)+10x1 = x0-expr.subs(var,x0).evalf(digits)/ \deriv.subs(var,x0).evalf(digits)

while abs(x1-x0)>tol:x0 = x1x1 = x0-expr.subs(var,x0).evalf(digits)/ \deriv.subs(var,x0).evalf(digits)

return x1


Symbolic Calculus


Newton Results

After importing this as nm, we call this as e.g.

nm.ntn_mth(sp.sin(x)-sp.Rational(3,4),x,0,.1e-25)nm.ntn_mth(sp.sin(x)-sp.Rational(3,4),x,3,.1e-25)

yielding 0.848062078981481008052944338998418080and 2.29353057460831223040969904428108480,respectively. Obviously we can just add integer multiples of 2πto get the rest.


Symbolic Calculus


Amusement

Incidentallynm.ntn_mth(x*x-.5,x,1,.1e-25)produces 1/

√2 to 24 digits:

0.707106781186547524400844362104849039

Curiously,sp.sqrt(.5).evalf(25)gave the answer 0.7071067811865475727373109. TheNewton method gave a different answer: more, or lessaccurate?

sp.sqrt(sp.Rational(1,2)).evalf(25)yielded 0.7071067811865475244008444.


Symbolic Calculus


Integration and Pretty Printing

We can antidifferentiate expressions.

expr=alpha*x*x+2*x+1sp.integrate(expr,x)gives alpha*x**3/3 + x**2 + x

If we are evaluating an integral, rather than an antiderivative,we need limits.

sp.integrate(expr,[x,0,x])produces alpha*x**3/3 + x**2 + x.sp.integrate(expr,(x,-1,x))yields alpha*x**3/3 + alpha/3 + x**2 + x.


Symbolic Calculus


Pretty Printing

In this situation, some better presentation of results would help.Sympy can “pretty print” results.

sp.pprint(sp.integrate(expr,x))3

alpha*x 2-------- + x + x

3

We can turn on “pretty printing” for all results using the function

sp.init_printing()


Symbolic Calculus


Integrate Anything

If the integral can’t be done using the Fundamental Theorem,then we can do it numerically.

sp.integrate(sp.sin(x**2+sp.exp(x)),(x,-1,2)))just returns the statement.

Butsp.integrate(sp.sin(x**2+sp.exp(x)),(x,-1,2)).evalf(20)gives 1.2536589539827120186.


Symbolic Calculus


Inert Functions

Often we want to “set up, but do not evaluate” integrals orderivatives or whatever. Sympy has special inert functions forthat.

a = sp.Integral(x**2,x)gives/

|| 2| x dx|/


Symbolic Calculus


Inert Evaluation

To evaluate an inert function, use the doit() method.

a.doit()results in3

x-3


Symbolic Calculus


Solving Equations

Sympy can solve systems of equations.

sp.solve(expr,x)gives

____________ / ____________ \\/ -alpha + 1 - 1 -\\/ -alpha + 1 + 1/

[------------------, --------------------]alpha alpha

Note that no equals signs are required. The understanding isthat the expression is to be equal to zero. Thus, to solve x2 = 5we would usesp.solve(x*x-5,x).


Symbolic Calculus


Solving Equations

... systems of equations.

sp.solve([expr,alpha+3],[x,alpha])produces [(-1/3, -3), (1, -3)]


Symbolic Calculus


Example

Find the value of x where 2x2 − x − 1 is minimized.

sp.solve(sp.diff(2*x**2-x-1,x),x)yields [1/4].


Symbolic Calculus


Example

Find the arc length of e−x2between -1 and 1.

integ = 1+sp.diff(sp.exp(-x*x),x)**2arclen=sp.integrate(integ,[x,-2,2]).evalf(20)produces 5.2518928987736186293.


Symbolic Calculus


Formal Derivatives

To solve differential equations, Sympy needs to be able tocreate variables that depend on other variables: i.e. formalfunctions.

y = sp.Function(’y’)makes y into a function depending on other unspecifiedvariables.


Symbolic Calculus


Derivatives

Once we have formal functions, we need to be able to takeformal derivatives of them.

yprime = sp.Derivative(y(t),t)producesd--(y(t))dt


Symbolic Calculus


Derivatives

Since the independent variables were unspecified, we can dowhat we need.

ysubt = sp.Derivative(y(t,x),t)yieldsd-(y(t, x))dt


Symbolic Calculus


Ordinary Differential Equations

Now we can solve DEs:

sp.dsolve(sp.Derivative(y(t),t)-y(t),y(t))results in

ty(t) = C1*e

sp.dsolve(sp.Derivative(y(t),t,2)+9*y(t),y(t))givesy(t) = C1*sin(3*t) + C2*cos(3*t)


Symbolic Calculus


Components of an Expression

Often we need to pick off pieces of an expression. Eachexpression has an args array containing the pieces.

expr = alpha*x*x+2*x+1expr.argsreturns (1,2*x,alpha*x**2).

Thus, if we want just the linear term, we can ask for

expr.args[1]which gives 2*x

Remember that 2*x is itself an expression, which means it hasan args method. Thus, expr.args[1].args[0] is 2.


Symbolic Calculus


Initial Value Problems

If we need to evaluate those constants, we can do it using theordinary solve() method. Here is one way (among many).We will solve y ′′ + 9y = 0, y(0) = 1, y ′(0) = −1.

eqn=sp.dsolve(sp.Derivative(y(t),t,2)+9*y(t),y(t))soln = eqn.args[1]This means soln = C1*sin(3*t) + C2*cos(3*t).

Now we just solvesp.solve(\[soln.subs(t,0)-1,diff(soln,t).subs(t,0)+1], \[C1,C2])which gives {C1: -1/3, C2: 1}


Symbolic Calculus


Zillions

There are a zillion more things we can do and know aboutSympy. Like all such packages, it is desperately complex whenyou get into it.

Still, I hope we have seen that some easy things to do areeasy. . .


Symbolic Calculus

k. cooper - washington state university

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