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# 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 1stKind given its point, X, and the order, N.
TI-86 Program tcheby
(127 bytes)
Prompt X,N
If X>1 : Goto A
If X
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
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