Correct and incorrect answer from linsolve

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Oddur Bjarnason
Oddur Bjarnason 2016-10-14
编辑： Walter Roberson 2016-10-17
The following gives a correct answer:
A=[-2/5,1/5;2/5,-1/5;1,1]
B=[0;0;1]
linsolve(A,B)
ans =0.3333 0.6667
The following, however, gives an incorrect answer:
C=[-0.2,0.3;0.25,-0.3;1,1]
D=[0;0;1]
C\D
ans = 0.5699 0.4297
The correct answer, found by substitution, is 0.6000 0.4000
How can I get a correct answer by linsolve?

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回答（1 个）

Walter Roberson
Walter Roberson 2016-10-14
x = sym('x',[2 1]);
C*x
ans =
(3*x2)/10 - x1/5
x1/4 - (3*x2)/10
x1 + x2
You have a system of 3 equations in 2 unknowns. It is overdetermined, and might not have any exact solution. The \ operation will do a least-squared fit to find an answer that is least bad in some sense.
Solving (3*x2)/10 - x1/5 = 0 for x1 gives x1 = (3*x2)/2. Substituting that back into C*x gives
0
(3*x2)/40
(5*x2)/2
solving (3*x2)/40 = 0 for x2 gives x2 = 0. Substituting that back gives (5*0)/2 = 1 which is 0 = 1 which has no solution.
>> C*[0.6;0.4]
ans =
0
0.03
1
So 0.6 0.4 is not a solution after-all.
>> sum((D-C*(C\D)).^2)
ans =
0.000407074042245239
>> sum((D-C*[0.6;0.4]).^2)
ans =
0.0009
so the solution found by C\D gives less of an error than [0.6 0.4] does.
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Steven Lord
Steven Lord 2016-10-17
Now the cause of the problem is clear. You wrote:
This yields the following system of linear equations:
0.8*SS1+0.3*SS2=SS1 0.25*SS1+0.7*SS2=SS2 SS1+SS2=1
You have a typo in your second equation. You wrote 0.25 instead of 0.2. So linsolve is solving the problem you told it to solve, not the problem you wanted it to solve. When I write the correct coefficient matrix, both linsolve and backslash (\) return the answer you expected.
A = [0.8-1 0.3; 0.2 0.7-1; 1 1];
b = [0; 0; 1];
x1 = linsolve(A, b)
x2 = A\b
Since you're interested in Markov chains you may find the PageRank and Markov Chains section of the chapter on Linear Equations in Cleve's Numerical Computing in MATLAB interesting and informative.
Oddur Bjarnason
Oddur Bjarnason 2016-10-17
Thank you Steven, When I was filling in the coefficient Matrix I read 0.2S1 to be 0.25. So you are right - it was a typo.

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