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

^{st}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 (1**

^{st}Kind)The program tcheby calculates the numerical value of the Chebyshev polynomials of the 1

^{st}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

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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