## Thursday, February 14, 2019

### TI-86: Angle Between Vectors, Vandermonde Matrix, Least Squares Algorithm, Chebyshev Polynomials (1st Kind)

TI-86: Angle Between Vectors, Vandermonde Matrix, Least Squares Algorithm, Chebyshev Polynomials (1st Kind)

The TI-86 has one of the best interfaces for a graphing calculator I ever had the joy to work with. The TI-86 is an update of the TI-85. Here are some programs for the TI-86.

Angle Between Vectors

The program vangle calculates the angle between two angles. The angle is calculated in degrees.

TI-86 Program vangle
(75 bytes)

Prompt V1
Prompt V2
Disp cos⁻¹ (dot(V1,V2)/(norm V1*V2))

Example: [2, -3, 4] and [8, 1, -2]
Angle: 83.5823268926°

Vandermonde Matrix

The program vander creates a matrix based on a list of coefficients.

Example: {x, y, z} produces the matrix

[ [x^0, x^1, x^2], [y^0, y^1, y^2], [z^0, z^1, z^2] ]

TI-86 Program vander
(125 bytes)

Input “List: “, L1
dimL L1 → N
{N, N} → dimL MA
For(R, 1, N)
For(C, 1, N)
L1(R)^(C-1) → MA(R,C)
End
End
Disp “MA=”
Pause MA

Example: {2, 4, 7}
Result: [ [1, 2, 4], [1, 4, 16], [1, 7, 49] ]

Least Square Algorithm

The program LSQ taxes the matrices X and Y (Y is a one-column matrix), and calculates
(X^T X)^-1 (X^T Y).

LSQ is used to fit statistical fits with least squares.

TI-86 Program LSQ
(118 bytes)

Disp “Least Squares”
Input “Matrix X: “,MX
Input “Matrix Y: “,MY
(MX^T * MX)^-1 * (MX^T * MY) → ML
Disp “ML= “
Pause ML

Example:
MX = [ [1, 3, 2.0], [1, 4, 2.3], [1, 5, 2.6], [1, 8, 2.9] ]
MY = [ [1.6], [1.8], [2.1], [2.3] ]

Results:
ML = [ [-0.14444444449], [-1.666666667E-2], [0.88888888889] ]

Chebyshev Polynomials (1st Kind)

The program tcheby calculates the numerical value of the Chebyshev polynomials of the 1st Kind given its point, X, and the order, N.

TI-86 Program tcheby
(127 bytes)

Prompt X,N
If X>1 : Goto A
If X<-1 :="" b="" goto="" span="">
cos(N * cos⁻¹ X) → A
Goto C
Lbl A
cosh(N * cosh⁻¹ X) → A
Goto C
Lbl B
(-1)^N * cosh(N * cosh⁻¹ X) → A
Lbl C
Disp A

Example:
X = -2.5, N = 4, Result: 263.5
X = 0.5, N = 4, Result: -0.5
X = 2.5, N = 4, Result: 263.5

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.

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