Library of High Speed Computation of Fractional Order Differentiation and Integration

High Speed Computation Programs of Fractional Integrals and Differentials
Paper from 'A high-speed algorithm for computation of fractional differentiation and fractional integration'

Library of High Speed Computation of

Fractional Order Differentiation and Integration

 

(MATLAB Function Version)

(C Language Version)


2016.6

MotionLabo, Inc.

 

  The libraries provided here include the programs that execute numerical calculations of fractional order differentiation and integration.

  The programs in the libraries, with the data compression based on the algorithm proposed by Fukunaga and Shimizu (2013), execute high-speed numerical computations of fractional order differentiation and integration of given functions, of numerical integration of fractional differential (and/or integral) equations that include fractional order derivatives (and/or fractional order integrations), and of data analysis based on the solutions of fractional differential (and/or integral) equations.

  The library is called to calculate fractional derivatives (and/or integrals) in a main program, in which a fractional differential (and/or integral) equation is solved. Several sample programs are provided for instruction of the library. Please refer the sample programs shown later.

The libraries are provided a type of “MATLAB version” and three types of “C language versions.”

   In the high-speed algorithm, the precision is controlled by five parameters, which include time step size h, and the order of approximation K. The precision and the computing cost appropriate for individual problems are determined by combination of the values of the parameters. We offer several type computation libraries suitable for the purposes of computation.

  In MATLAB-A1 and C-A1, which are given Table 1 below, other parameters than h are all set as default value. This simplifies the precision setting. The precision is controlled by h as in usual numerical integration for differential equations. The highest precision is achieved when h is taken about 1/100 of characteristic time of function change. (The characteristic time, or the dynamical time, is defined as the time in which a bounded function changes its value by its amplitude. As for the sine function, sin (kt), the dynamical time is 1/k.)  MATLAB-A1 and C-A1 types are suited for computation of fractional differentiation of functions, simulation of small sized problems with several variables, data analysis, etc.

  For computations with higher precision, the higher order of approximation is necessary. In C-B1, the order K=3 is taken so that computations with higher precision can be achieved comparing to C-A1. The maximal accuracy as a whole is obtained when h satisfies h=Td/1000, where Td is the dynamical time of the function in consideration. In addition to h, the parameter M is provided in order to control the accuracy of calculation. At a given h, the best accuracy and cost is obtained whenanother  parameter M is chosen to satisfy a certain condition.C-B1 can compute problems of middle sized simulations and problems covered by C-A1.

  In C-C1, all the parameters except for K=4 can be chosen in hand to control the precision of calculation of fractional derivatives. In programming, the parameters can be chosen freely, which provides best precision suited for individual purpose with minimal computing cost. With the order K=4, the highest precision can be obtained which is achievable in the double- precision programming. C-C1 is suited for large sized simulations and the problems covered by C-B1 . 

 

Table 1 Performance list (* note)

Product

Category No.

Application fields

K

Step no.

Minimal Error

On sale

MATLAB Standard

M-A1

Introductory and small sized simulation

2

40,960

3×105

In preparation

Standard

C-A1

Introductory and small sized simulation

2

163,840

3×105

Coming soon

High Grade

C-B1

Middle sized simulation

3

163,840

3 107

Available Now 

Professional

C-C1

Large sized simulation

4

Unlimited

3×1011

In preparation

Fortran Professional

F-C1

In preparation

 

(*note in Table 1) 

- MATLAB Standard program is offered as a .p function type. In C language, the programs are offered in the form of lib (for Windows) or .a (for Mac).

- In MATLAB Standard program, a number of variables fractionally differentiated is less and equal to 3. In C language programs, there is no limitation on a number of variables fractionally differentiated.

- The relative errors in the table are rough estimate for fractional derivative of a single function. The precision is dependent on type of differentiations, type of individual functions, or type of simultaneous equations.

- The maximal steps for C-B1 tabulated in Table 1 is for larger h. For small h with large M(>40), the maximal step increases with M for M =< 400

- In the C language programs, we provide optional libraries in which multiple-order fractional derivatives and integrals can be calculated.

- In the C-C1 library, the three fractional orders can be calculated as the standard option.

- Fortran language programs are now under developing.

 

Table2 Calculation speed (M = 40)

steps

CA1 [s]

CB1 [s]

CC1 [s]

FC1 [s]

General Method [s]

1000

0.0012

0.00105

0.00105

0.00089

0.00143

2000

0.0030

0.0030

0.0026

0.00207

0.0055

5000

0.0089

0.0091

0.0073

0.0061

0.033

10,000

0.0197

0.0181

0.0157

0.0138

0.132

20,000

0.043

0.041

0.032

0.030

0.52

50,000

0.115

0.120

0.092

0.086

3.3

100,000

0.255

0.264

0.193

0.188

15.1

1,000,000

 

 

2.24

2.17

(1500)

(Operating environment:Mac OSX version 10.9.5, cpu: 3.1 GHz, memory: 16GB)


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Price:  For one license per one PC

 

 MATLAB Standard (M-A1): US$ 105 (includes Japanese tax)     In preparation

 C Standard (C-A1):          US$ 315 (includes Japanese tax)     Coming Soon!

 C High Grade (C-B1):       US$ 525 (includes Japanese tax)     Available Now  

 

C Professional (C-C1):      US$ 2,520 (includes Japanese tax)    In preparation

 

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Operating environment

Windows version

  Windows7sp1 32bit, 62bit

  CPU 2.4GHz or higher

  Memory 2GB or higher

  Microsoft Visual Studio 2015(for C Language Programs)

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Mac version

  Mac OSX  version 10.9.5

  CPU 2.4GHz or higher

  Memory 2GB or higher

  gcc version 4.9.2

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For users’ reference, four typical program examples are given in the following link. Please click the corresponding link.

“Short course on fractional derivatives and integrals” is given in the following link.

Short course on fractional derivatives and integrals .pdf,  click here

 

Inquiry and Order

Flow from order to receipt:

(1) You will pay the fee into order account.   C-B1: 

(2) After checking the payment in the account, we will start to publish a license for your computer. We need “Host Name” and “Physical Address (for Local Area Connection)” of your PC for publishing the license.

(3) The ordered programs and related documents will be sent to you.

(4) You will check the operation of the program you bought on your computer.

 

About inquiry:

If you have any questions on the products which you bought, you can ask to following.

 

ml.eng_00@motionlabo.com

 

 

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