.
So we have
iteration number 2.
The initial guess . . . not the initial guess, I should say, the
estimate which we have now for x1, x2,
and x3 is given as follows,
and that's from the previous iteration, that's what we got for the value of x1,
x2, and x3.|And now we're going to again calculate our x1.|So x1
is 1, minus 3 x2,
plus 5 x3, divided by 12.
And so I'll get 1, minus 3 times x2, which is 4.9,
plus 5 times 3.0923,
divided by 12,
and this number here turns out to be 0.14679.
Same thing I'm going to do for x2.
x2 will be, formula is 28, minus x1,
minus 3 x3, divided by 5.
So this turns out to be 28, minus x1, again you use the most recent value,
the value here is 0.5, but we're going to use the most recent value for x1,
which is 0.14679, minus 3 times
x3, which is 3.0923,
divided by 5, and this number
here turns out to be 3.7153.
We're going to do this same thing for x3.
The formula for x3 which we wrote was 76, minus 3 x1,
minus 7 x2, divided by 13.
So this becomes 76, minus 3 times x1, so we're going to use the most recent
value again here, is 0.14679, minus 7
times x2, again, we're going to use the most recent value which we obtained, which is
3.7153, divided by 13,
and this number here turns out to be equal to 3.8118.
So that's what we're getting as the value for x3.
So what that means is that the . . . at the end of the second iteration,
this is what we're getting the value for x1, x2, and x3.
We are getting the value as
0.14679,
3.7153, and 3.8118,
and that's what we're getting as our solution at the end of the
second iteration.|Now what we're going to do is, again, we have to calculate
the absolute relative approximate errors.|So,
our new x1, x2, x3, which I just wrote
just a little while ago, is 0.14679,
3.7153, and
3.8118, and the old guess will be the one which we obtained
at the end of the first iteration, or these are the values we
used when we started this iteration, 0.5,
4.9, and 3.0923.
And if you go ahead and calculate the absolute relative
approximate errors, this is what you're going to get.|I'm going to leave this as a
homework exercise for you, you're going to get 240.61
percent, the absolute relative approximate error for
x2 will turn out to be 31.889 percent,
and the absolute relative approximate error for the third
will be 18.874 percent.
So again, these are basically calculated from the present guess and the previous guess.
Now, if you're going to find the maximum of these absolute relative approximate errors,
240.61, 31.889,
and 18.874, you are going to get
240.61 percent.
So what you are finding out here is that at the end of the first iteration, this number,
the maximum absolute relative approximate error was 100 percent, and now you're getting 240.61
percent.|So it might lead you do believe that this process
is diverging, but that's not the case.|So if I
go ahead and write down here, in a simple form, the iteration number
and the maximum absolute
relative approximate error, in percentage.|We already know that
at the end of the first iteration, the maximum value was 100.|At 2, it is
240.6.|At 3, it turns
out to be 80.23.|So it increased,
but now it's decreasing, and all the way up to six, if I go up to six, the maximum value
which I get for the relative approximate error is 0.743
percent.|So it does converge.|So it might start to diverge
a little bit here, but eventually it's going to converge.|In fact, if you conduct more iterations, this number
is going to get smaller and smaller.|At the end of the sixth iteration
.
the value for x1, x2, and x3 which you get
is 0.99919,
3.0001, and
4.0001.|This is what you get as the solution
at the end of the sixth iteration for x1, x2, and x3.|And, in fact,
the exact value, if you would calculate it by just using some
programming language or computational package like MATLAB, or Mathcad,
or Mathematica, or Maple, you will get 1, 3, and 4,
and you can very well see that it's pretty close to what we're getting at the end of the sixth iteration.
Of course, we can always calculate the relative, absolute relative true error to see that how close it
is to quantify that error, but at the same time, you can see that if you conduct more
iterations, you are going to get closer and closer to the exact value, so it does converge, although
it might give you the impression that it's starting to diverge right there.|And that's
the end of this segment.
.