Eddie's Math and Calculator Blog

Saturday, October 19, 2024

Swiss Micros DM32: Reimann-Louiville Fractional Integral of x^p

Swiss Micros DM32: Reimann-Louiville Fractional Integral of x^p

Introduction

The program presented today calculates the Riemann-Louiville integral of:

f(t) = t^p, where p is a real number.

The formula for this integral is:

cDx^(-v) = 1 / Γ(v) * ∫( (x – t) * t^p dt, t = c, t = x)

= ∫( ((x – t) * t^p) / (v -1)! dt, t = c, t = x)

I covered these type of integrals on my September 14, 2024 blog.

DM32 Program: Reimann-Louiville Fractional Integral of x^p

LBL F

INPUT C

INPUT X

INPUT V

INPUT P

FN= I

RCL C

RCL X

∫ FN d T

RTN

LBL I

RCL X

RCL- T

RCL V

y^x

RCL T

RCL P

y^x

RCL V

RTN

Here is a text version that can be transferred to a dm32 state file (fractionalintegralm.d32):

https://drive.google.com/file/d/1E-wUq4GW5dX06VZ-5WWwy7KRm3SN_uyq/view?usp=sharing

Examples

Run program F: XEQ F. Make sure that V > 0.

C	X	V	P	Result (FIX 5)
0	5	2	2	52.08333
1	6	3	3	385.41667
2	7	1.5	3	717.69103
0	1	1.75	4	0.05298

Source

Kimeu, Joseph M., "Fractional Calculus: Definitions and Applications" (2009).Masters Theses & Specialist Projects. Paper 115. http://digitalcommons.wku.edu/theses/115

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Saturday, October 12, 2024

Casio fx-CG 50: Centroid of a 2D spaces

Casio fx-CG 50: Centroid of a 2D spaces

The program CENTROID calculates the center point for an area covered by:

y(x) ≥ 0, x ≥ lower limit, x ≤ upper limit

The center point is calculated by:

x-center = ∫( x * y(x) dx, x = lower limit, x = upper limit) / A

y-center = ∫(1 / 2 * y(x)^2 dx, x = lower limit, x = upper limit) / A

where A = ∫( y(x) dx, x = lower limit, x = upper limit)

Casio fx-CG 50 Program Code: CENTROID

Here is the code, which includes a graphic representation of y(x) and the location of the centroid. In this code, the Y is bold and it comes from the VARS menu. Do not merely use ALPHA+Y. For calculators with monochrome screens, such as the fx-9750G/fx-9860G series, leave out the color commands (Black, Blue, Red). The program assumes that there no preset plots or more than one function to be plotted.

This program works best for functions y(x) ≥ 0 for x ∈ [lower limit, upper limit].

Here is a text-only version:

Examples

Example 1: y = x^2 , lower limit = 0, upper limit = 1

X-Center = 3 / 4 = 0.75

Y-Center = 3 / 10 = 0.3

Example 2: y = 2 * cos( x / 2 ), lower limit = 0, upper limit = π

X-Center ≈ 1.141592654

Y-Center ≈ 0.7853981634

Example 3: y = 3, lower limit = 1, upper limit = 5

X-Center = 3

Y-Center = 1.5

Example 4: y = -4 * x^2 + 2 * x + 6, lower limit = -0.5, upper limit = 1

X-Center = 1 / 4

Y-Center = 307 / 110

Eddie

Saturday, October 5, 2024

Sum of Sequential Integers (featuring Swiss Micros DM32)

Sum of Sequential Integers (featuring Swiss Micros DM32)

What is the sum of the following:

100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999?

It’s doable on a calculator, but going straight forward will require a lot of keystrokes (unless you have access to the sigma function (Σ)).

Note that:

100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999

= 100 + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + …. + (100 + 897) + (100 + 898) + (100 + 899)

= (100 + 0) + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + …. + (100 + 897) + (100 + 898) + (100 + 899)

Notice the sum starts with a base, 100. The sequence goes for 900 terms in the form of 100+n where n = 0 to 899. Let’s look at a general case.

Let x be a base integer, and S be the sum as:

S = (x + 0) + (x + 1) + (x + 2) + (x + 3) + …. + (x + n)

Rearranging the terms leads us to:

S = (x + x + x + x + … + x) + (0 + 1 + 2 + 3 + 4 + … + n)

There are n + 1 pairs:

S = (x * (n + 1)) + (0 + 1 + 2 + 3 + 4 + … + n)

S = (x * (n + 1)) + (1 + 2 + 3 + 4 + … + n)

The sum of Σ( k, k = 1 to k = n) = 1 + 2 + 3 + 4 + … + n = n * (n + 1) / 2

Then:

S = (x * (n + 1)) + n * (n + 1) / 2

Let’s look at some specific examples:

n = 1

S = (x + 0) + (x + 1)

S = (x + x) + (0 +1)

S = 2 * x + 1

n = 2

S = (x + 0) + (x + 1) + (x + 2)

S = (x + x + x) + (0 + 1 + 2)

S = 3 * x + 3

2 * 3 / 2 = 3

n = 3

S = (x + 0) + (x + 1) + (x + 2) + (x + 3)

S = (x + x + x + x) + (0 + 1 + 2 + 3)

S = 4 * x + 6

3 * 4 / 2 = 6

S = 500 + 501 +502 + 503

S = (500 + 0) + (500 + 1) + (500 + 2) + (500 + 3)

S = (500 + 500 + 500 + 500) + (0 + 1 + 2 + 3)

S = ((3 + 1) * 500) + (3 * 4 / 2)

S = 2000 + 6

S = 2006

S = 250 + 251 + 252 + 253 + … + 269 + 270

Base = 250

n = 270 – 250 = 20

Then:

S = 250 * (20 + 1) + 20 * 21 / 2

S = 5460

Let’s go our original problem:

S = 100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999

Base = 100

n = 999 -100 = 899

Then:

S = 100 * (899 + 1) + 899 * 900 / 2

S = 90000 + 404550

S = 494550

The program code calculates the sum:

Swiss Micros DM32 Program (also HP 32SII)

S01 LBL S

S02 INPUT X

S03 INPUT N

S04 RCL N

S05 1

S06 +

S07 RCL× N

S08 RCL N

S09 1

S10 RCL+ N

S11 ×

S12 2

S13 ÷

S14 +

S15 RTN

Eddie

Sunday, September 29, 2024

TI-30Xa Algorithm: Synthetic Division

Introduction

Synthetic division is a fairly simple division algorithm that divides a polynomial by the nominal (x-a):

p(x) / (x - c) where

p(x) = a_n * x^n + a_n-1 * x^(n-1) + a_n-2 * x^(n - 2) + … + a_1 * x + a_0

c = a numerical constant

the result, q(x), is a polynomial of order n-1:

q(x) = b_n-1 * x^(n-1) + b_n-2 * x^(n - 2) + … + b_1 * x + b_0 + b_r / (x - c)

b_r = remainder term

where

b_n-1 = a_n

b_i-1 = b_i * c + a_i for i = n-2 down to r (“-1”, see the example below)

For example, for the polynomial:

p(x) = a3 * x^3 + a2 * x^2 + a1 * x + a0

q(x) = p(x) / (x - c)

p(x) has the order n = 3, so q(x) will have the order n = 3 - 1 = 2:

q(x) = b2 * x^2 + b1 * x + b0 + br / (x - c)

b2 = a3

b1 = b2 * c + a2

b0 = b1 * c + a1

br = b0 * c + a0 (the algorithm stops, this is the remainder term)

If br = 0, the x - c divides p(x) evenly, and x = c is a root of p(x).

Note: for any “missing” terms, fill the term with 0.

Example: x^3 + 4 * x - 5 becomes x^3 + 0 * x^2 + 4 * x - 5.

Calculator Algorithm

Store c in one of the memory registers 1, 2, or 3. We’ll call this memory register m for the purpose of the blog.
Note the first coefficient of q(x): a_n
Compute the rest of the coefficients as follows: [ × ] [ RCL ] m [ + ] a_ni [ = ]

Examples

Remember: m is memory register 1, 2, or 3, your choice.

Example 1: (21 * x^2 + 42 * x + 144) / (x - 12)

c = 12

a2 = 21

a1 = 42

a0 = 144

b1 = a2 = 21

12 [ STO ] m

Enter 21

b0:

[ × ] [ RCL ] m [ + ] 42 [ = ]

Result: 294

br:

[ × ] [ RCL ] m [ + ] 144 [ = ]

Result: 3672

Stop.

q(x) = 21 * x + 294 + 3672 / (x - 12)

Example 2: (2 * x^3 + x - 3) / (x - 3) = (2 * x^3 + 0 * x^2 + x - 3) / (x - 3)

c = 3

a3 = 2

a2 = 0

a1 = 1

a0 = -3

b2 = a3 = 2

3 [ STO ] m

Enter 2

b1:

[ × ] [ RCL ] m [ + ] 0 [ = ]

Result: 6

b0:

[ × ] [ RCL ] m [ + ] 1 [ = ]

Result: 19

br:

[ × ] [ RCL ] m [ + ] 3 [ +/- ] [ = ]

Result: 54

q(x) = 2 * x^2 + 6 * x + 19 + 54 / (x - 3)

Example 3: (4 * x^3 + 8 * x^2 - 5 * x + 3) / (x + 2)

c = -2

a3 = 4

a2 = 8

a1 = -5

a0 = 3

b2 = a3 = 4

2 [ +/- ] [ STO ] m

Enter 4

b2:

[ × ] [ RCL ] m [ + ] 8 [ = ]

Result: 0

b3:

[ × ] [ RCL ] m [ + ] 5 [+/-] [ = ]

Result: -5

br:

[ × ] [ RCL ] m [ + ] 3 [ = ]

Result: 13

q(x) = 4 * x^2 - 5 + 13 / (x + 2)

I think this is a fundamental algorithm for students and math enthusiasts to learn, and it’s fairly simple to get a hang of it.

Until next time,

Eddie

Quick update: Starting October 5, 2024, my schedule will allow me to blog once a week. Regular posts will go live every Saturday. Thank you for your support and compliments. I wish you all well.

Saturday, September 28, 2024

TI-84 CE and TI-86: Matrix Row Statistics

TI-84 CE and TI-86: Matrix Row Statistics

Introduction

The program MROWST will take a matrix and a desired row and calculate three statistics points:

* Sum of the row

* Mean of the row

* Difference Vector: row – mean for each element

TI-84 CE Program MROWST

Matrices used: [ J ] (entry matrix), [ I ] (difference vector)

Input “MATRIX? “, [ J ]

Input “ROW? “, R

dim([ J ])

Ans(2) → D

0 → S

For(I, 1, D)

S + [ J ](R, I) → S

End

S / D → M

{1, D} → dim([ I ])

For(I, 1, D)

[ J ](R, I) – M → [ I ](1, I)

End

ClrHome

Disp “SUM: “ + toString(S)

Disp “MEAN: “ + toString(M)

Disp “DIFF VECTOR:”

Pause [ I ]

TI-86 Program MROWST

Matrices used: MJ (entry matrix), MD (difference vector), mone (vector of ones)

Input “Matrix? “, MJ

Input “Row? “, R

dim MJ

Ans(2) → D

D → dim mone

Fill(1, mone)

dot(mone, MJ(R)) → S

S / dim mone → M

MJ(R) - (M * mone) → MD

ClLCD

Disp “Sum:”, S

Disp “Mean:”, M

Disp “Diff Vector:”

Pause MD

Example

Matrix:

[ [ -9, -4, -1, -9, 4 ]

[ -9, -7, -4, -4, 9 ]

[ 7, 1, -9, -7, 8 ] ]

Row 1:

Sum: -1

Mean: -0.2

Difference Vector: [ 9.2, -3.8, -0.8, -8.8, 4.2 ]

Row 2:

Sum: -15

Mean: -3

Difference Vector: [ -6, -4, -1, -1, 12 ]

Row 3:

Sum: 0

Mean: 0

Difference Vector: [ 7, 1, -9, -7, 8 ]

Source:

Stuerke, Cecil. “Demonstration of Principal Component Analysis on TI-86” IEEE 2008

Eddie

Quick update: Starting October 5, 2024, my schedule will allow me to blog once a week. Regular posts will go live every Saturday. Thank you for your support and compliments. I wish you all well.