# University of Illinois at Urbana-Champaign
# MATH-484 - Fall 2013
# Instructor Bernard Lidicky
#
# From lecture 35 - showing some basic use of sage
# Sage can be downloaded at http://www.sagemath.org
# Or there is an online version of sage at http://www.sagenb.org

# Help:
# More details can be found at:
# http://www.sagemath.org/doc/constructions/linear_algebra.html
# http://www.sagemath.org/doc/reference/calculus/sage/calculus/calculus.html
#

# The following commands can be just paster into sagenb.org
# I want to solve Ax = b

# QQ stands for rational numbers. RR for reals
M = MatrixSpace(QQ,3,3) # Something for generating matrices

A = M([ 1,2,3, 4,5,6, 7,8,10]) # creates a matrix

print "A=",A  # prints
view(A) # nice print


V = VectorSpace(QQ,3) # vector space... over Q
b = V([4,5,6])


A.inverse()
A*A.inverse()

x = A.inverse()*b

A*x

float(10/3)  # how to write as a number and not as fraction

octave.solve_linear_system(A,b)

# Change to RR

var('x,y')    # say that x,y will be varibles
f(x,y)=x^3+2*y^2 # function of two variables
f.diff()  # gradient
H = f.diff(2) # hessian
H(x=0,y=-1/2) # hessian at 0,-1/2
H(x=0,y=-1/2).eigenvalues() # eigenvalues...