Run a file:
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%runfile filename.sage |
Create matrices:
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matrix(RR, nrows, ncols) # real ring matrix(SR, nrows, ncols) # symbolic ring matrix([[a,b,c],[d,e,f],[g,h,i]]) # real vector([a,b,c]) # single column matrix |
Solve M * X = Y (where M – co-efficients; Y – constants) :
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M.solve_right(Y) |
Create variables (symbolic math):
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x,y = var('x,y') tan_theta = y / x; |
Here are some examples of Sage used as a Computer algebra system.
Define the symbols x and y using var. Then define r2 in terms of x and y as per the Pythagorean theorem.
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sage: x, y = var('x, y') sage: r2 = x * x + y * y sage: r2 |
The output remains in terms of x and y:
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x^2 + y^2 |
This works with trigonometry…
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sage: r = sqrt(r2) sage: sin0 = y/r sage: cos0 = x/r sage: theta = atan(sin0/cos0) sage: tan(theta) |
Output:
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y/x |
…and matrices too. Below is an example of a matrix to translate and rotate a point in two dimensions.
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sage: dx, dy, r = var('dx, dy, r') sage: s = sin(r) sage: c = cos(r) sage: tx = matrix([[c, s, dx], ....: [-s, c, dy], ....: [0, 0, 1]]) sage: v = vector([x, y, 1]) sage: tx * v |
Output:
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(x*cos(r) + y*sin(r) + dx, y*cos(r) - x*sin(r) + dy, 1) |
Sage also does differentiation and integration but I am still trying to wrap my head around those two concepts.
Given the set of equations:
\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 15 \end{bmatrix} \times \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right] = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right] \)
Below is how to solve them using Sage:
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sage: A = matrix([[5,0,0],[0,10,0],[0,0,15]]) sage: Y = vector([1,2,3]) sage: X = A.solve_right(Y) sage: X |
The output:
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(1/5, 1/5, 1/5) |
In order to get the output as floating point numbers use the numerical_approx function:
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sage: n(X, digits=4) |
The output:
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1 |
(0.2000, 0.2000, 0.2000) |