% Story problem: 2 lbs carrots and 3lbs apples cost $10.15. 1 lb carrots and
% 7 lb apples cost $17.12. How much do carrots and apples cost per pound?
% eqns are
% 2c + 3a = 10.15
% 1c + 7a = 17.12
% To solve in Matlab, write it in matrix-vector form as an Ax=b problem.
% A holds the coefficients of the unknowns in the left-hand sideof the eqn
% x is the vector of unknowns c and a
% b is the vector of known right-hand-side values 10.15 and 17.12
A = [2 3 ; 1 7]
A =
2 3
1 7
b = [10.15 ; 17.12]
% Solve Ax=b for x using Matlab's "backslash" operator
x = A\b
x =
1.7900
2.1900
% Verify that the solution satisfies the original equations by using
% long-winded high-school arithmetic. First assign the components of
% x into variables c and a.
c = x(1)
c =
1.7900
a = x(2)
a =
2.1900
% verify that 2c + 3a = 10.15
2*c + 3*a
ans =
10.1500
% verify that c + 7a = 17.12
c + 7*a
ans =
17.1200
% Now verify the solution using matrix multiplication
% ie verify that Ax = b
A
A =
2 3
1 7
x
x =
1.7900
2.1900
A*x
ans =
10.1500
17.1200
% New problem
% 1c + 2a + 3t = 18.84
% 2c + 1a = 5.97
% 6a + 2t = 23.52
% Enter this into matlab as an Ax=b problem
A = [ 1 2 3 ; 2 1 0 ; 0 6 2]
A =
1 2 3
2 1 0
0 6 2
b = [18.84 ; 5.97 ; 23.52]
b =
18.8400
5.9700
23.5200
% Solve Ax=b using the backslash operator
x = A\b
x =
1.6900
2.5900
3.9900
% So the answer is
% carrots cost $1.69/lb
% apples cost $2.59/lb
% tomatoes cost $3.99/lb
% Wasn't that a total breeze, compared to solving three equations in three
% unknowns by hand? Hooray for numerical linear algebra!
% Verify first equation long-windedly
c = x(1)
c =
1.6900
a = x(2)
a =
2.5900
t = x(3)
t =
3.9900
1*c + 2*a + 3*t
ans =
18.8400
% Now verify all equations by multiplying A*x and comparingto b
A*x
ans =
18.8400
5.9700
23.5200
A
A =
1 2 3
2 1 0
0 6 2
x
x =
1.6900
2.5900
3.9900
A*x
ans =
18.8400
5.9700
23.5200
b
b =
18.8400
5.9700
23.5200
% Observe above that A*x equals b. Can also verify Ax=b by computing
% Ax-b and seeing that it's zero
% that
A*x-b
ans =
1.0e-14 *
0.3553
0
0
% Note that Ax-b is not exactly zero. It has magnitude 1e-14, so it's
% very nearly zero. That's because the numerical computations are done
% in finite precision (there's rounding error).
% Question: how do you do subscripts in Matlab?
A
A =
1 2 3
2 1 0
0 6 2
% access the i,j = 3,2 element of A via A(3,2)
A(3,2)
ans =
6
% Get second column of A
A(:,2)
ans =
2
1
6
% In matlab, A(:,j) means all rows in jth col
% In matlab, A(i,:) means all cols in ith row
A
A =
1 2 3
2 1 0
0 6 2
A(1,:)
ans =
1 2 3
A(2,:)
ans =
2 1 0
A(3,:)
ans =
0 6 2
A
A =
1 2 3
2 1 0
0 6 2
% Change the 3,1 elem to 99
A(3,1) = 99
A =
1 2 3
2 1 0
99 6 2
% Change the 3rd row to 7, 2, 19
A(3,:) = [7 2 19]
A =
1 2 3
2 1 0
7 2 19
% Change the 2nd column to -1 4 6
A(:,2) = [-1 ; 4 ; 6]
A =
1 -1 3
2 4 0
7 6 19