- functions
- Taylor Series in Python
In this post, we will review how to create a Taylor Series with Python and for loops. Then we will refactor the Taylor Series into functions and compare the output of our Taylor Series functions to functions from Python's Standard Library.
A Taylor Series is an infinite series of mathematical terms that when summed together approximate a mathematical function. A Taylor Series can be used to approximate $e^x$, and $cosine$.
An example of a Taylor Series that approximates $e^x$ is below.
$$ {e^x} \approx \sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{n!}}} \approx 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} \ + \ ... $$
We can see that each term in the Taylor Series expansion is dependent on that term's place in the series. Below is a chart that shows each term of the Taylor Series in a row. The columns of the table represent the term index, the mathematical term and, how to code that term in Python. Note that the factorial() function is part of the math module in Python's Standard Library.
| Term Index | Mathematical Term | Term coded in Python |
|---|---|---|
| 0 | $x^0/0!$ | x**0/math.factorial(0) |
| 1 | $x^1/1!$ | x**1/math.factorial(1) |
| 2 | $x^2/2!$ | x**2/math.factorial(2) |
| 3 | $x^3/3!$ | x**3/math.factorial(3) |
| 4 | $x^4/4!$ | x**4/math.factorial(4) |
Code the Taylor Series by writing out each term individually¶
We can combine these terms in a line of Python code to estimate $e^2$. The code below calculates the sum of the first five terms of the Taylor Series expansion of $e^x$, where $x=2$. Note the math module needs to be imported before math.factorial() can be used.
In[1]:
import mathx = 2e_to_2 = x**0/math.factorial(0) + x**1/math.factorial(1) + x**2/math.factorial(2) + x**3/math.factorial(3) + x**4/math.factorial(4)print(e_to_2)
7.0
Our Taylor Series approximation of $e^2$ was calculated as 7.0. Let's compare our Taylor Series approximation to Python's math.exp() function. Python's math.exp() function raises $e$ to any power. In our case, we want to use math.exp(2) because we want to calculate $e^2$.
In[2]:
print(math.exp(2))
7.38905609893065
Our Taylor Series approximation 7.0 is not that far off the calculated value 7.389056... using Python's exp() function.
Use a for loop to calculate a Taylor Series¶
If we want to get closer to the value of $e^x$, we need to add more terms to our Taylor Series. The problem is coding each individual term is time-consuming and repetitive. Instead of coding each term individually, we can use a for loop. A for loop is a repetition structure in Python that runs a section of code a specified number of times. The syntax for coding a for loop in Python using the range() function is below:
for <var> in range(<num>): <code>Where <var> is any valid Python variable name and <num> is an integer to determines how many times the <code> in the loop runs.
We can recreate our approximation of $e^2$ with 5 terms using a for loop. Note we need to set the variable e_to_2 to 0 before the loop starts. The mathematical operator += in the line e_to_2 += x**i/math.factorial(i) is equivalent to e_to_2 = e_to_2 + x**i/math.factorial(i).
In[3]:
import mathx = 2e_to_2 = 0for i in range(5): e_to_2 += x**i/math.factorial(i) print(e_to_2)
7.0
The result 7.0 is the same as the result we calculated when we wrote out each term of the Taylor Series individually.
An advantage of using a for loop is that we can easily increase the number of terms. If we increase the number of times the for loop runs, we increase the number of terms in the Taylor Series expansion. Let's try 10 terms. Note how the line for i in range(10): now includes 10 passed to the range() function.
In[4]:
import mathx = 2e_to_2 = 0for i in range(10): e_to_2 += x**i/math.factorial(i) print(e_to_2)
7.3887125220458545
The result is 7.38871.... Let's see how close that is to $e^2$ calculated with Python's exp() function.
In[5]:
print(math.exp(2))
7.38905609893065
The result is 7.38905.... We get closer to the value of $e^2$ when 10 terms are used in the Taylor Series compared to when 5 terms were used in the Taylor Series.
Refactor the for loop into a function¶
Next, let's refactor the code above that contained a for loop (which calculated $e^2$) into a function that can compute $e$ raised to any power estimated with any number of terms in the Taylor Series. The general syntax to define a function in Python is below.
def <func name>(<arg 1>, <arg 2>, ...): <code> return <output>Where def is the Python keyword that defines a function, <func name> is a valid Python variable name, and <arg 1>, <arg 2> are the input arguments passed to the function. <code> that runs when the function is called must be indented (the standard indentation is 4 spaces). The keyword return denotes the <output> of the function.
Let's code our for loop that approximates $e^2$ into a function.
In[6]:
import mathdef func_e_to_2(n): x = 2 e_to_2 = 0 for i in range(n): e_to_2 += x**i/math.factorial(i) return e_to_2
If we call our function func_e_to_2() with the input argument 10, the result is the same as when we ran the for loop 10 times.
In[7]:
out = func_e_to_2(10)print(out)
7.3887125220458545
The output of our func_e_to_2() function is 7.38871....
We can make our function more general by setting x (the number that $e$ gets raised to) as an input argument. Note how now there are two input arguments in the function definition (x, n). x is the number $e$ is raised to, and n is the number of terms in the Taylor Series (which is the number of times the for loop runs on the inside of the function definition).
In[8]:
import mathdef func_e(x, n): e_approx = 0 for i in range(n): e_approx += x**i/math.factorial(i) return e_approx
Let's calculate $e^2$ using 10 terms with our new func_e() function.
In[9]:
out = func_e(2,10)print(out)
7.3887125220458545
The result is 7.38871..., the same result as before.
An advantage to writing our Taylor Series expansion in a function is that now the Taylor Series approximation calculation is reusable and can be called in one line of code. For instance, we can estimate the value of $e^5$ with 10 terms, by calling our func_e() function with the input arguments (5,10).
In[10]:
out = func_e(5,10)print(out)
143.68945656966488
The result is 143.68945.... Let's see how close this value is to Python's exp() function when we make the same calculation $e^5$.
In[11]:
out = math.exp(5)print(out)
148.4131591025766
The result is 148.41315.... The Taylor Series approximation calculated by our func_e() function is pretty close to the value calculated by Python's exp() function.
Use a for loop to calculate the difference between the Taylor Series expansion and Python's exp() function¶
Now let's use a for loop to calculate the difference between the Taylor Series expansion as calculated by our func_e() function compared to Python's exp() function. We'll calculate the difference between the two functions when we use between 1 and 10 terms in the Taylor Series expansion.
The code below uses f-strings, which is a Python syntax for inserting the value of a variable in a string. The general syntax for an f-string in Python is below.
f'string statment {<var>}'Where f denotes the beginning of an f-string, the f-string is surrounded by quotes ' ', and the variable <var> is enclosed in curly braces { }. The value of <var> will be printed out without the curly braces.
In[12]:
import mathx = 5for i in range(1,11): e_approx = func_e(x,i) e_exp = math.exp(x) e_error = abs(e_approx - e_exp) print(f'{i} terms: Taylor Series approx= {e_approx}, exp calc= {e_exp}, error = {e_error}')
1 terms: Taylor Series approx= 1.0, exp calc= 148.4131591025766, error = 147.41315910257662 terms: Taylor Series approx= 6.0, exp calc= 148.4131591025766, error = 142.41315910257663 terms: Taylor Series approx= 18.5, exp calc= 148.4131591025766, error = 129.91315910257664 terms: Taylor Series approx= 39.33333333333333, exp calc= 148.4131591025766, error = 109.079825769243275 terms: Taylor Series approx= 65.375, exp calc= 148.4131591025766, error = 83.03815910257666 terms: Taylor Series approx= 91.41666666666667, exp calc= 148.4131591025766, error = 56.996492435909937 terms: Taylor Series approx= 113.11805555555556, exp calc= 148.4131591025766, error = 35.295103547021048 terms: Taylor Series approx= 128.61904761904762, exp calc= 148.4131591025766, error = 19.794111483528989 terms: Taylor Series approx= 138.30716765873015, exp calc= 148.4131591025766, error = 10.10599144384644910 terms: Taylor Series approx= 143.68945656966488, exp calc= 148.4131591025766, error = 4.723702532911716
Note how the error decreases as we add terms to the Taylor Series. When the Taylor Series only has 1 term, the error is 147.41.... When 10 terms are used in the Taylor Series, the error goes down to 4.7237....
Use a break statement to exit a for loop early.¶
How many terms would it take to produce an error of less than 1? We can use a break statement to drop out of the for loop when the error is less than 1. The code below calculates how many terms are needed in the Taylor Series, when $e^5$ is calculated, to keep the error less than 1.
In[13]:
import mathx = 5for i in range(1,20): e_approx = func_e(x,i) e_exp = math.exp(x) e_error = abs(e_approx - e_exp) if e_error < 1: break print(f'{i} terms: Taylor Series approx= {e_approx}, exp calc= {e_exp}, error = {e_error}')
12 terms: Taylor Series approx= 147.60384850489015, exp calc= 148.4131591025766, error = 0.8093105976864479
The output shows that it takes 12 terms in the Taylor Series to drop the error below 1.
Create a function to estimate the value of $cos(x)$ using a Taylor Series¶
Next, let's calculate the value of the cosine function using a Taylor Series. The Taylor Series expansion for $\cos(x)$ is below.
$$ \cos \left( x \right) \approx \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}\frac{{{x^{2n}}}}{{\left( {2n} \right)!}}} \approx 1 - \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} - \frac{{{x^6}}}{{6!}} + \frac{{{x^8}}}{{8!}} - \frac{{{x^{10}}}}{{10!}} \ + \ ... $$
We can code this formula into a function that contains a for loop. Note the variable x is the value we are trying to find the cosine of, the variable n is the number of terms in the Taylor Series, and the variable i is the loop index which is also the Taylor Series term number. We are using a separate variable for the coefficient coef which is equal to $(-1)^i$, the numerator num which is equal to $x^{2i}$ and the denominator denom which is equal to $(2i!)$. Breaking the Taylor Series formula into three parts can cut down on coding errors.
In[14]:
import mathdef func_cos(x, n): cos_approx = 0 for i in range(n): coef = (-1)**i num = x**(2*i) denom = math.factorial(2*i) cos_approx += ( coef ) * ( (num)/(denom) ) return cos_approx
Let's use our func_cos() function to estimate the cosine of 45 degrees. Note that func_cos() function computes the cosine of an angle in radians. If we want to calculate the cosine of 45 degrees using our function, we first have to convert 45 degrees into radians. Luckily, Python's math module has a function called radians() that makes the angle conversion early.
In[15]:
angle_rad = (math.radians(45))out = func_cos(angle_rad,5)print(out)
0.7071068056832942
Using our func_cos() function and 5 terms in the Taylor Series approximation, we estimate the cosine of 45 degrees is 0.707106805.... Let's check our func_cos() function compared to Python's cos() function from the math module.
In[16]:
out = math.cos(angle_rad)print(out)
0.7071067811865476
Using Python's cos() function, the cosine of 45 degrees returns 0.707106781... This value is very close to the approximation calculated using our func_cos() function.
Build a plot to compare the Taylor Series approximation to Python's cos() function¶
In the last part of this post, we are going to build a plot that shows how the Taylor Series approximation calculated by our func_cos() function compares to Python's cos() function.
The idea is to make a plot that has one line for Python's cos() function and lines for the Taylor Series approximation based on different numbers of terms.
For instance, if we use 3 terms in the Taylor Series approximation, our plot has two lines. One line for Python's cos() function and one line for our func_cos() function with three terms in the Taylor series approximation. We'll calculate the cosine using both functions for angles between $-2\pi$ radians and $2\pi$ radians.
In[17]:
import mathimport numpy as npimport matplotlib.pyplot as plt# if using a Jupyter notebook, include:%matplotlib inlineangles = np.arange(-2*np.pi,2*np.pi,0.1)p_cos = np.cos(angles)t_cos = [func_cos(angle,3) for angle in angles]fig, ax = plt.subplots()ax.plot(angles,p_cos)ax.plot(angles,t_cos)ax.set_ylim([-5,5])ax.legend(['cos() function','Taylor Series - 3 terms'])plt.show()
We can use a for loop to see how much better adding additional terms to our Taylor Series approximation compares to Python's cos() function.
In[18]:
import mathimport numpy as npimport matplotlib.pyplot as plt# if using a Jupyter notebook, include:%matplotlib inlineangles = np.arange(-2*np.pi,2*np.pi,0.1)p_cos = np.cos(angles)fig, ax = plt.subplots()ax.plot(angles,p_cos)# add lines for between 1 and 6 terms in the Taylor Seriesfor i in range(1,6): t_cos = [func_cos(angle,i) for angle in angles] ax.plot(angles,t_cos)ax.set_ylim([-7,4])# set up legendlegend_lst = ['cos() function']for i in range(1,6): legend_lst.append(f'Taylor Series - {i} terms')ax.legend(legend_lst, loc=3)plt.show()
We see the Taylor Series with 5 terms (the brown line) comes closest to approximating Python's cos() function. The Taylor Series with 5 terms is a good approximation of the cosine of angles between about $-\pi$ and $\pi$ radians. The Taylor Series with 5 terms is a worse approximation for angles less than $-\pi$ or greater than $\pi$. As the angle gets further away from zero radians, the estimate of the cosine using a Taylor Series gets worse and worse.
Summary¶
In this post, we reviewed how to construct the Taylor Series of $e^x$ and $\cos(x)$ in Python. First, we coded each term of the Taylor series individually. Next, we used a for loop to calculate $n$ terms in the Taylor Series expansion. We then refactored the code with the for loop into a function and parameterized our function so that we could calculate $e$ raised to any number calculated out to any number of terms. At the end of the post, we coded the Taylor Series of $\cos(x)$ into a Python function. Finally, we used our Taylor Series cosine function to build a plot with Matplotlib that shows how the Taylor Series approximation compares to Python's cos() function for angles between $-2\pi$ and $2\pi$ radians.
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FAQs
How do you solve Taylor series in Python? ›
Use a for loop to calculate a Taylor Series
The mathematical operator += in the line e_to_2 += x**i/math. factorial(i) is equivalent to e_to_2 = e_to_2 + x**i/math. factorial(i) . The result 7.0 is the same as the result we calculated when we wrote out each term of the Taylor Series individually.
The Taylor formula is the key. It gives us an equation for the polynomial expansion for every smooth function f. However, while the intuition behind it is simple, the actual formula is not. It can be pretty daunting for beginners, and even experts have a hard time remembering if they haven't seen it for a while.
How do you verify Taylor's theorem? ›Taylor's Series Theorem
Assume that if f(x) be a real or composite function, which is a differentiable function of a neighbourhood number that is also real or composite. Then, the Taylor series describes the following power series : f ( x ) = f ( a ) f ′ ( a ) 1 ! ( x − a ) + f ” ( a ) 2 !
This is the Taylor polynomial of degree n about 0 (also called the Maclaurin series of degree n). More generally, if f has n+1 continuous derivatives at x=a, the Taylor series of degree n about a is n∑k=0f(k)(a)k! (x−a)k=f(a)+f′(a)(x−a)+f”(a)2! (x−a)2+…
How do you do a Taylor series step by step? ›- Step 1: Calculate the first few derivatives of f(x). We see in the formula, f(a). ...
- Step 2: Evaluate the function and its derivatives at x = a. ...
- Step 3: Fill in the right-hand side of the Taylor series expression. ...
- Step 4: Write the result using a summation.
Solve the initial value problem y' = -2xy2, y(0) = 1 for y at x = 1 with step length 0.2 using Taylor series method of order four. Using Taylor series method of order four solve the initial value problem y' = (x - y)/2, on [0, 3] with y(0) = 1. Compare solutions for h = 1, 1/2, 1/4 and 1/8.
Which method is better than Taylors method? ›Taylor Series and Euler methods
For increased accuracy we can apply more than one corrector step. To accelerate the convergence, Newton's method is recommended.
A Taylor Series is an expansion of some function into an infinite sum of terms, where each term has a larger exponent like x, x2, x3, etc.
Do calculators use Taylor series? ›Calculators don't actually use the Taylor series but the CORDIC algorithm to find values of trigonometric functions. The Cordic algorithm is based on thinking of the angle as the phase of a complex number in the complex plane, and then rotating the complex number by multiplying it by a succession of constant values.
What type of error does Taylor series have *? ›The Lagrange error bound of a Taylor polynomial gives the worst-case scenario for the difference between the estimated value of the function as provided by the Taylor polynomial and the actual value of the function.
How do you tell if a Taylor polynomial is increasing or decreasing? ›
If f (x0) = 0, the parabola f (x0) + T2(x) has its vertex at x0, and you will recognize the familiar rule to determine max- ima and minima. If f (x0) > 0, the parabola is increasing, and hence it is concave up, while if f (x0) < 0 it will be concave down.
What does Taylor series tell us? ›The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.
How many best methods of Taylor are there? ›Answer: In Taylor's technique, there are four techniques: 1. Functional Foremanship 2. Standardisation and Simplification of Work 3. Work-Study 4.
Why is Taylor's theorem useful? ›Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions.
How do you know if a Taylor series converges? ›If L=0, then the Taylor series converges on (−\infty, \infty). If L is infinite, then the Taylor series converges only at x=a.
How do you write Taylor series expansion? ›Taylor's Series Theorem Statement:
Assume that if f(x) be a real or composite function, which is a differentiable function of a neighbourhood number that is also real or composite. Then, the Taylor series describes the following power series: f(x)=f(a)f′(a)1! (x−a)+fp(a)2!
A Taylor series can be used to describe any function ƒ(x) that is a smooth function (or, in mathematical terms, "infinitely differentiable.") The function ƒ can be either real or complex. The Taylor series is then used to describe what the function looks like in the neighborhood of some number a.
How do you find the Taylor series of two variables? ›Taylor's formula for functions of two variables , up to second derivatives. g(0) + tg'(0) + t2 2 g '' (0 ) , and if t is small and the second derivative is continuous, g(t) 7 g(0) + tg'(0) + t2 2 g''(0). f (x,y) 7 f (a,b) + d f d x (a,b)(x - a) + d f d y (a,b)(y - b).
How is Taylor series used in deep learning? ›Use cases of Taylor series in deep learning
Taylor approximation can be applied to isolate the difficulties like shattered gradients, in the training of neural networks. Tylor expansion and approximation can be used for investigating the hypothesis that adaptive optimizers converge to better solutions.
We finally use the derived Taylor method to derive second order Runge Kutta method of order two on time scales and carry out examples using it. AMS subject classification: Keywords: Time scales — Dynamic Equations — Taylor approximation — Runge Kutta.
What is the special advantage of Runge-Kutta method over Taylor series method? ›
R.K Methods do not require prior calculation of higher derivatives of y(x) ,as the Taylor method does. Since the differential equations using in applications are often complicated, the calculation of derivatives may be difficult.
Why is improved Euler method better than Euler method? ›The Improved Euler's Method addressed these problems by finding the average of the slope based on the initial point and the slope of the new point, which will give an average point to estimate the value. It also decreases the errors that Euler's Method would have.
What is the difference between Taylor and Maclaurin series? ›The Taylor Series, or Taylor Polynomial, is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. A Maclaurin Polynomial, is a special case of the Taylor Polynomial, that uses zero as our single point.
Does Taylor series always converge? ›The function may not be infinitely differentiable, so the Taylor series may not even be defined. The derivatives of f(x) at x=a may grow so quickly that the Taylor series may not converge. The series may converge to something other than f(x).
Did NASA use calculators? ›They did not have computers or calculators. What did they use? A slide rule! It was used to help find answers to hard questions.
Do engineers use Taylor series? ›Taylor series have wide reaching applications across mathematics, physics, engineering and other sciences. And the concept of approximating a function, or data, using a series of function is a fundamental tool in modern science and in use in data analysis, cell phones, differential equations, etc..
Why do we study Taylor series? ›Probably the most important application of Taylor series is to use their partial sums to approximate functions. These partial sums are (finite) polynomials and are easy to compute.
Is the Taylor series always equal the function? ›We can see by this that a function is equal to its Taylor series if its remainder converges to 0; i.e., if a function f can be differentiated infinitely many times, and limn→∞Rn(x)=0, then f is equal to its Taylor series.
What is order of accuracy in Taylor series? ›Definition: The power of Δx with which the truncation error tends to zero is called the Order of Accuracy of the Finite Difference approximation. The Taylor Series Expansions: FD and BD are both first order or are O(Δx) (Big-O Notation) CD is second order or are O(Δx2) (Big-O Notation)
How accurate is Taylor series? ›Taylor's Theorem guarantees such an estimate will be accurate to within about 0.00000565 over the whole interval [0.9,1.1] .
How do you find the degree 3 of a Taylor polynomial? ›
In the third degree of Taylor polynomial, the polynomial consists of the first four terms ranging from 0 to 3. =f(a)+f′(a)(x−a)+f″(a)2(x−a)2+f″(a)6(x−a)3. Therefore, ⇒f′(x)=1x,f″(x)=−1x2,f‴(x)=2x3.
How do you know if an interval is increasing decreasing or constant? ›Step 1: A function is increasing if the y values continuously increase as the x values increase. Find the region where the graph goes up from left to right. Use the interval notation. Step 2: A function is decreasing if the y values continuously decrease as the x values increase.
How do you use the factorial function in Python? ›- # Python program to find.
- # factorial of given number.
- import math.
- def fact(n):
- return(math.factorial(n))
- num = int(input("Enter the number:"))
- f = fact(num)
- print("Factorial of", num, "is", f)
It's a tie!") elif user_action == "rock": if computer_action == "scissors": print("Rock smashes scissors! You win!") else: print("Paper covers rock! You lose.") elif user_action == "paper": if computer_action == "rock": print("Paper covers rock!
What is Taylor series in programming? ›C Programming Mathematics: Exercise-24 with Solution
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. Example: The Taylor series for any polynomial is the polynomial itself.
*= Multiply AND. It multiplies right operand with the left operand and assign the result to left operand. c *= a is equivalent to c = c * a. /= Divide AND. It divides left operand with the right operand and assign the result to left operand.
How do you calculate factorial easily? ›- Determine the number. Determine the number you are finding the factorial of. ...
- Write the sequence. Using the factorial formula, you can write out the sequence of numbers that you'll multiply. ...
- Multiply the numbers. Once you have the sequence of numbers written down, you can multiply them together.
The == operator compares the value or equality of two objects, whereas the Python is operator checks whether two variables point to the same object in memory. In the vast majority of cases, this means you should use the equality operators == and !=
What is Elif in Python? ›In Python, elif is short for "else if" and is used when the first if statement isn't true, but you want to check for another condition. Meaning, if statements pair up with elif and else statements to perform a series of checks. A full if/elif/else block is in the code example below.
What does Elif mean in Python? ›The elif keyword is used in conditional statements (if statements), and is short for else if.