For details, see How to Use.
Vnano  Compute Integrated Value Numerically (Numerical Integration)
An example program (script) to perform integration numerically. This script uses simpson's rule by default, but you can use the rectangular method and the trapezoidal method with modifying a line in code.
Code of this script is written in the Vnano. Code written in C, C++, and Java® are also distributed in Code section. In addition, the Vnano is a subset of the VCSSL, so you can execute this script as VCSSL code, with changing the extention of the script file to ".vcssl" (In such case, please enable the "import" statement at the near the top of the code, which is commentedout by default).
Also, if you want to plot the graph of the integrated function, see the next article and code.
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How to Use
Download and Extract
At first, click the "Download" button at the above of the title of this page by your PC (not smartphone). A ZIP file will be downloaded.
Then, please extract the ZIP file. On general environment (Windows®, major Linux distributions, etc.), you can extract the ZIP file by selecting "Extract All" and so on from rightclicking menu.
» If the extraction of the downloaded ZIP file is stopped with security warning messages...
Execute this Program
Next, open the extracted folder and execute this VCSSL program.
For Windows
Doubleclick the following batch file to execute:
For Linux, etc.
Execute "VCSSL.jar" on the commandline terminal as follows:
java jar VCSSL.jar
» If the error message about nonavailability of "java" command is output...
How to Use After Executing The Program
Output the Integrated Value
When the program has started, a window will be launched. Then, when the calculation has completed, a small window will be launched, and the integrated value will be displayed on it.
By default, the calculation will be completed soon. If you make the value of "N" in the script larger, required time of the calculation become longer,
How to Change the Integrant Function, Interval, etc.
For changing the integrant function, modify the content of the function "f(x)" defined in the script. For changing the interval of the integration, modify values of variables "A" and "B" in the script. In addition, if necessary, modify the value of the variable "N" in the script, which is the number of tiny intervals (See: Code section).
By default settings, "cos(x)" will be integrated from x = 0 to x = 1.
How to Change The Algorithm of the Integration
In this script, as algorithms of the numerical integration, you can use the rectangular method, the trapezoidal method, and the Simpson's rule. Each algorithm is implemented as a line in the script (See: Code section), so please add/remove "//" (a line starts with "//" will be ignored) at those lines, for selecting the algorithm you want to use. By default, the Simpson's rule will be used.
Other  How to Plot The Graph of The Integrated Function
If you want to plot the graph of the integrated function, we are destributing the modified version of this script for such purpose:
In the above article, we are explaining how to plot the output data into a graph.
Code and Algorithms
Entier Code
Code of this program is written in the Vnano. The Vnano has a simple Clike syntax, so if you accustomed with C or Clike programming languages, you probably can read/customize the content of code easily.
Let's see the entier code:
The Top of Code
Code starts with the following content:
The first line "coding ..." is for declaring the textencoding of the script file, for preventing encoding/decodingrelated probrems (e.g.: "mojibake").
The next line "import ..." (which is used for loading a library) is necessary if you execute this script as VCSSL code, but it is not necessary when you exeute this script as Vnano code, so it is commentedout by default.
Definitions of The Integrant Function, The Interval of The Integration, etc.
In the next part, the integrant function, the interval of the integration, and so on are defined:
When you want to customize settings of the integration, modify the above code. Basically, define the integrant function as the function "f(x)" in the above code, and input the value of the lower/upperlimit of the integration to the variable "A" / "B".
"N" represents the number of tiny intervals (for details, see the latter sections). When you make the value of N larger, the precision of the integrated value basically increses, but required time for the calculation become longer. Please note that, if you set the immediately large value to N, the precision might decrease, because the number of total cycles of the integration loop become very large, so tiny numerical errors affect many times to the result.
The Core Part: Performing The Numerical Integration
The next part is, the core part of this script performing the numerical integration:
If you read the above code without any background knowledge, you might not be able to understand meanings of comments in the above code, but don't worry. Let's see keypoints of numerical integration algorithms briefly and graphically.
Calculate The Value of The Integration By CountingUp Area Numerically
There are many kind of numerical integration algorithms in the world, but three methods used in this script are based on a common concept: calculate the value of the integration by numerically countingup area between the integrant function f(x) and xaxis:
The area painted by the red color in the above figure equals to the integrated value of f(x) from a to b.
Why? What condition is assumed? It is difficult to answer those questions strictly. It requires some complicated mathematical discussions about definitions of integral and integrability.
We don't want to discuss about it here, so, Let's assume that we choose a standard (not trickey, リーマン integrable) function as the integrant function f(x), and then the area of the above figure equals to the integrated value.
Rectangular Method
As mentioned above, our goal is very simple: we want to calculate the area painted by the red color in the above figure, as precise as possible. Let's discuss how calculate the area in programs.
Probably the most simple way is as follows: Firstly split the entier area into N"tiny interval"s, and then approximate the value of area of each tiny interval as a rectangle of which height is the value of f(x) at the leftbottom of the tiny area, and finally calculate the summation value of area of all rectangles (see the following figure).
This is socalled "the rectangular method".
Does it make sense? Let's return to code. You can use the rectangular method by removing "//" from the head of the line of the rectangular method, and adding "//" to the head of the line of the Simpson's rule. Then, with removing unnecessary lines, code become as follows:
In the above code, the lefthandside of the last line "f(x) * delta" represents area of a rectangle of a tiny interval. The above code calculates the summation value of it of all rectangles with moving the value of x in the forloop.
By the way, by default settings, this script calculated the integrated value of cos(x) from 0 to 1, so the theoretically correct value is sin(1) = 0.8414709848078965... . On the other hand, the numerically integrated value depends on the value of N. The followings are results by using the rectangular method:
Result under N=10: 0.8637545267950129
Result under N=100: 0.8437624610086617
Result under N=1000: 0.8417007635323798
Result under N=10000: 0.8414939689913762
Result under N=100000: 0.8414732832893704
As the above, with making the value of N larger, we can get more precise result. More strictly, when we make the value of N about 1 digit (10 times) larger, the length of correct digits in the result increases about 1 digit.
Trapezoidal Method
Errors between the correct value and calculated values by using the rectangular method are mainly caused by that: we has approximated the value of area between the xaxis and the "roundy curve" of f(x) by using "edgy rectangles".
Hence, as the next step, Let's approximate the curve of f(x) by using a polyline of which nodes are \( (x,y) = (x_i, f(x_i)) \), where \(x_i\) is the xcoordinate value of the leftbottom point of ith tiny interval (i = 0, 1, 2, ..., N1) :
By the above improvement of the approximation, rectangles of the rectangular method become trapezoides. This is socalled "the trapezoidal method".
You can use the trapezoidal method by removing "//" from the head of the line of the trapezoidal method, and adding "//" to the head of the line of the Simpson's rule. Then, with removing unnecessary lines, code become as follows:
In the above code, the lefthandside of the last line "( f(x) + f(x+delta) ) * delta / 2.0" represents area of a trapezoid of a tiny interval. The above code calculates the summation value of it of all trapezoides with moving the value of x in the forloop.
Calculation results by using the trapezoidal method, under the same settings with the rectangular method, are followings:
Result under N=10: 0.8407696420884198
Result under N=100: 0.8414639725380026
Result under N=1000: 0.8414709146853138
Result under N=10000: 0.841470984106668
Result under N=100000: 0.8414709848008824
As the above, when we make the value of N about 1 digit (10 times) larger, the length of correct digits in the result increases about 2 digits. In the contrast, when we used the rectangular method, we were required to make the value of N about 100 times larger, for improving the precision about 2 digits.
Simpson's Rule
Let's continue to improve the approximation. In the trapezoidal method, we have approximated the curve of f(x) in a tiny interval by using a line, as depicted in the following figure:
As commented in the above figure, in a tiny interval, if we can approximate f(x) by using a manually integrable curve which fits with f(x) well, its integrated value gives more good approximation of area of a tiny interval, than a trapezoid.
Hence, Let's approximate f(x) in a tiny interval by using a quadratic function throughing same points as f(x), at bothside boundaries and the center of the tiny interval:
In general, parameters of a ndimentional polynomial equation can be determined when values at (n + 1) points are given, so parameters of the above quadratic (2dimentional) function can be determined, because values at (2 + 1) points are given. Then, we can calculate the integrated value of the determined quadratic function manually.
The result of the above calculation is wellknown as the Simpson's rule. By using the Simpson's rule, we can get the approximate value of the area of ith tiny interval, \(A_i\), as follows:
\[ A_{i} = \frac{ f(x_i) + f(x_i + \Delta x) + 4f(x_i+ \Delta x / 2) }{6} \Delta x \]
As the rectangular method and trapezoidal method, calculate the summation of \(A_i\) for all i ( = 0, 1, 2, ..., N1).
This script uses the Simpson's rule by default. With removing unnecessary lines, code calculating area by using the Simpson's rule are as follows:
In the above code, the lefthandside of the last line "( f(x) + f(x+delta) + 4.0 * f(x+delta/2.0) ) * delta / 6.0" represents an approximate value of area of a tiny interval \(A_i\). The above code calculates the summation value of it of all \(A_i\) with moving the value of \(x (= x_i)\) in the forloop.
Let's check the precision:
Result under N=10: 0.8417720922382719
Result under N=100: 0.8414710140343371
Result under N=1000: 0.8414709848108186
Result under N=10000: 0.8414709848078956 (Near the limit of 64bit float)
Result under N=100000: 0.8414709848078982 (Near the limit of 64bit float)
As the above, when we make the value of N about 1 digit (10 times) larger, the length of correct digits in the result increases about 4 digits. In the contrast, when we used the rectangular method, we were required to make the value of N about 10000 times larger(!!!), for improving the precision about 4 digits. Even when we used the trapezoidal method, it required to make N 100 times larger.
Also, as we can recognize in above results, please note that we can NOT improve the precision of the result beyond the limit of the numerical data type used for calculations (in this script/article, we have used "64bit binaly floatingpoint number"). If you forget the above thing, and if you set the value of N to an immediately large number, the precision goes lower than the best, because the number of total cycles of the integration loop become very large, so tiny numerical errors affect many times to the result. The best value of N depends on f(x) and the interval of the integration, so check the behaviour of results with changing the value of N, as we did in this article.
License
The license of this VCSSL / Vnano code (the file with the extension ".vcssl" / ".vnano") is CC0 (Public Domain), so you can customize / divert / redistribute this VCSSL code freely.
Also, if example code written in C/C++/Java are displayed/distributed on Code section of this page, they also are distributed under CC0, unless otherwise noted.
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