Thursday, February 7, 2019

Casio fx-3650P: Chebyshev Polynomials, Roots of a Complex Number, Reversion Value, Mach Speed, Trapezoid Perimeter


Casio fx-3650P: Chebyshev Polynomials, Roots of a Complex Number, Reversion Value, Mach Speed, Trapezoid Perimeter

Chebyshev Polynomials of the First Kind

Values of the Chebyshev polynomials can calculated by the following:

T_n (x):

cos (n acos x) if |x| ≤ 1

cosh (n acosh x) if x ≥ 1

(-1)^n cosh (n acosh(-x)) if x ≤ 1

Inputs: x, y (n) order. The program assumes y is an integer.

Program (74 bytes):

? → X : ? → Y : X > 1 ⇒ Goto 1 :
-1 > X ⇒ Goto 2 : cos ( Y cos¹ X ) → A : Goto 3 :
Lbl 1 : cosh ( Y cosh¹ X ) → A : Goto 3
Lbl 2 : (-1)^Y * cosh ( Y cosh¹ X ) → A :
Lbl 3: A

Examples

T_Y(X)

X: -3, Y: 5, Result: -3363
X: 0.3, Y: 5, Result: 0.99888
X: 3, Y: 5, Result: 3363

Roots of a Complex Number

This program calculates the roots of a complex number (x + yi)^a, which is calculated as:

r^(1/n) * ( cos((θ + 2 π k)/n) + i*sin((θ + 2 π k)/n) )

where k = 0 to n-1
r =√(x^2 + y^2)
θ = angle(x + yi) = atan(y/x)

Program (66 bytes):

? → X : ? → Y : ? → A : 0 → M :
Rad : Pol(X,Y) : A x√ X → B :
Y → D : Lbl 1 : M ⊿ Rec(B, (D + 2 π M )/A) ⊿
Y ⊿ 1 M+ : M ≠ A ⇒ Goto 1 : M

The program stops when M = A.

Example:

(-2 + 4i)^(1/4)

X: -2, Y: 4, A: 4

(Fix 4 setting)
M: 0, X: 1.2701, Y: 0.7082
M: 1, X: -0.7082, Y: 1.2701
M: 2, X: -1.2701, Y: -0.7082
M: 3, X: 0.7082, Y: -1.2701
M: 4 (end)

Business: Reversion Value

Input:

A = n: number of payments
B = I%: periodic interest rate (in decimal)
C = monthly rent
D = asking price

Note: For monthly payments, divide the annual interest rate by 1200.

Program (55 bytes):

Fix 2 : ? → A : ? → B : ? → C : ? → D :
-C (1 + B) ( (1 + B)^A -1 ) ÷ B → X :
X + D (1 + B)^A → X : X

Example:
5 years of monthly payments, A = 60
Interest rate, B = 6/1200
Monthly Rate, C = 1895
Asking Price, D = 250000


Result: 204337.26

Source:
Roger F. Farish and Elbert B. Greynolds, Jr., Ph.D., CPA Calculator Analysis for Business and Finance Texas Instruments Incorporated 1978 ISBN 0-07-063757-1

Mach Speed

This program calculates the mach speed, a ratio of the speed of sound given:

A = the altitude of the object, such as a vehicle (in feet)
B = temperature (°F)
C = velocity of the vehicle (mph)

Equations used:

Mach Number = velocity of the vehicle / speed of sound

where Speed of Sound = 44100 / 1519 * √( 459.7 + B – 3.57 * A / 1000)

Program (52 bytes):

? → A : ? → B : ? → C :
44100 ÷ 1519 * √ ( 459.7 + B – 3.57 A ÷ 1000 ) → D :
C ÷ D → M

A lapse rate of -3.57° per 1000 ft altitude is assumed.

Example:

Input:
A: altitude = 2500 ft
B: temperature = 63 °F
C: velocity = 795 mph

Result:
mach speed: 1.208091501

Source:
Mach Number and Airspeed vs Altitude” Granite Island Group (Technical Surveillance Counter Measures) http://www.tscm.com/mach-as.pdf Retrieved January 26, 2019

Perimeter of a Trapezoid



P + A + B + C * (1/sin X + 1/sin Y)

Program (38 bytes):

Deg : ? → A : ? → B : ? → C : ? → X : ? → Y :
A + B + C ( 1 ÷ sin X + 1 ÷ sin Y )

Example:
A = 227.6
B = 305.4
C = 10.5
X = 86°
Y = 5°

Result: 663.999629

Eddie

All original content copyright, © 2011-2019. 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.