Algebraic Equations/Symbolic Tools in Visual Studio .NET

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CHAPTER 5 Algebraic Equations/Symbolic Tools
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We can enter the system as a set of character strings:
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>> >> >> >> eq1 eq2 eq3 eq4 = = = = 'w + '2*w 'w + 'w x + 4*y + 3*z = 5'; +3*x+y 2*z = 1'; 2*x 5*y+4*z = 3'; 3*z = 9';
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Now we call solve, passing the equations as a comma-delimited list:
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>> s = solve(eq1,eq2,eq3,eq4);
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We can then extract the value of each variable which solves the system of equations:
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>> w = sw w = 1404/127 >> x = sx x = 818/127 >> y = sy y = 53/127 >> z = sz z = 87/127
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Expanding and Collecting Equations
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In elementary school we learned how to expand equations For instance: (x + 2) (x 3) = x2 x 6
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MATLAB Demysti ed
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We can use MATLAB to accomplish this sort of task by calling the expand command Using expand is relatively easy For example:
>> expand((x 1)*(x+4))
When this command is executed, we get:
ans = x^2 +3*x 4
The expand function can be applied in other ways For example, we can apply it to trig functions, generating some famous trig identities:
>> expand(cos(x+y))
This gives us:
ans = cos(x)*cos(y) sin(x)*sin(y)
>> expand(sin(x y)) ans = sin(x)*cos(y) cos(x)*sin(y)
To work with many symbolic functions, you must tell MATLAB that your variable is symbolic For example, if we type:
>> expand((y 2)*(y+8))
MATLAB returns:
Unde ned function or variable 'y
To get around this, rst enter:
>> syms y
Then we get:
>> expand((y 2)*(y+8)) ans = y^2+6*y 16
CHAPTER 5 Algebraic Equations/Symbolic Tools
MATLAB also lets us go the other way, collecting and simplifying equations First let s see how to use the collect command One way you can use it is for distribution of multiplication Consider: x(x2 2) = x3 2x To do this in MATLAB, we just write:
>> collect(x*(x^2 2)) ans = x^3 2*x
Another example (with a new symbolic variable t):
>> syms t >> collect((t+3)*sin(t)) ans = sin(t)*t+3*sin(t)
Another algebraic task we can do symbolically is factoring To show that: x2 y2 = (x + y)(x y) Using MATLAB, we type:
>> factor(x^2 y^2) ans = (x y)*(x+y)
We can factor multiple equations with a single command:
>> factor([x^2 y^2, x^3+y^3])
Finally, we can use the simplify command This command can be used to divide polynomials For instance, we can show that (x2 + 9)(x2 9) = x4 81 by writing:
>> simplify((x^4 81)/(x^2 9)) ans = x^2+9
Here is another example Consider that: e2log3x = elog9x = 9x2
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We could have found this out with the MATLAB command:
>> simplify(exp(2*log(3*x))) ans = 9*x^2
The simplify command is useful for obtaining trig identities Examples:
>> simplify(cos(x)^2 sin(x)^2) ans = 2*cos(x)^2 1 >> simplify(cos(x)^2+sin(x)^2) ans = 1
Solving with Exponential and Log Functions
So far for the most part we have only looked at polynomials The symbolic solver can also be used with exponential and logarithmic functions First let s consider and equation with logarithms EXAMPLE 5-4 Find a value of x that satis es: log10 (x) log10 (x 3) = 1 SOLUTION 5-4 Base ten logarithms can be calculated by or represented by the log10 function in MATLAB So we can enter the equation thus:
>> eq = 'log10(x) log10(x 3) = 1';
CHAPTER 5 Algebraic Equations/Symbolic Tools
Then we call solve:
>> s = solve(eq);
In this case, MATLAB only returns one solution:
>> s(1) ans = 10/3
Now let s consider some equations involving variables as exponents Suppose we are asked to solve the system: y=3 x y=5 +1 We can call solve to nd the solution:
>> s = solve('y = 3^2*x','y = 5^x+1')
MATLAB returns:
s = x: [2x1 sym] y: [2x1 sym]
So it appears there are two values of each variable that solve the equation We can get the values of x out by typing:
>> sx(1) ans = 1/9*exp( lambertw( 1/9*log(5)*5^(1/9))+1/9*log(5))+1/9 >> sx(2) ans = 1/9*exp( lambertw( 1, 1/9*log(5)*5^(1/9))+1/9*log(5))+1/9
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These might not be too useful to the average man or woman So let s convert them to numerical values:
>> a = double(sx(1)) a = 02876 >> b = double(sx(2)) b = 16214
We can also extract the y values:
>> c = double(sy(1)) c = 25887 >> d = double(sy(2)) d = 145924
Do the results make sense We check The idea is to see if sx(1) satis es the rst equation giving sy(1) and ditto for the second array elements We nd:
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