CS 107 - Machine Problem 4

Solving Multivariable Algebraic Equations using Fractions

Due: 4/7/2003 at 2:00 pm

Write-up for an alternate (and simplier) algorithm for validating the series of equations.

For this assignment, you are to solve a series of N algebraic equations of N variables. The following is an example of this where N = 3.

     2a +  3b +  1c = 11
     4a + -2b +  3c =  9
     3a +  5c + -3c =  4

In solving the above series of equations we need to translate the series into the form:

     1a + 0b + 0c = v1
     0a + 1b + 0c = v2
     0a + 0b + 1c = v3

Thus showing the solution is a = v1, b = v2 and c = v3. The translation requires each equation to have the coefficient of one of the variables changed to 1 and the remaining coefficients change to 0. Each equation in the final series will have a different variable's coefficient changed to 1.

To change the coefficient of a variable to 1, we just need to divide all of the coeffients and the result of the equation by the current coefficient of the variable that we wish to have a coefficient of 1. For example, take the equation:

     2a +  3b +  1c = 11

To set the coefficient of the variable a to 1, divide each coefficient and the result of the equation by 2 to get:

     1a +  3/2b +  1/2c = 11/2

This equation would replace the original equation in the series of equations.

To zero out the coefficient of a variable in one equation, we must use a second equation whose coefficent for that variable is 1. This second equation is created by the process mentioned above. (Note: You can use an equation whose coefficient is not 1, but it requires a division by this coefficient. So you might as well do the division before this step. One fact is that the equation cannot have a coefficient of zero. This would cause a division by zero.) Once the two equations to be used have been determined, multiple the second equation (whose coefficient for that variable is 1) by the coefficient for that variable from the first equation. When multiplying the equation by a value, all coefficients and the result must be mutliplied by the value. Then subtract the equation obtained by the multiplication from the first equation. When subtracting, subtract the coefficients of like variables and subtract the results separately to obtain a new equation. This new equation should have a coefficient of zero for the intended variable and is to replace the first equation in the series of equations. For example, we want to zero out the coefficient of a in the following equation (to be referred to as the first equation).

     4a + -2b + 3c = 9

and we have the second equation of:

     1a + 3/2b + 1/2c = 11/2

When multiplying the second equation by 4 (the coefficient of a in the first equation), we get:

     4a + 6b + 2c = 22

This equation is subtracted from the first equation to get:

     0a + -8b + c = -13

This final equation would replace the equation of 4a +-2b +3c = 9 in the series of equations.

Your program is to solve a series of algebraic equations that may have up to 10 variables (with the same number of equations). Your program is to prompt the user for a filename that will contain the information for the series of equations. The first line of the file will contain a single integer value which informs the program of the number N for the series. This is the number of variable in each equation and the number of equations in the series. The next N lines in the file will contain N+1 integer values that correspond to the coefficients and the result for each equation. The last value on the line will be the result of the equation. An example data file is:

     3
     2   3   1   11
     4  -2   3    9
     3   5  -3    4

This data file would correspond to the following series of equations:

     2a +  3b +  1c = 11
     4a + -2b +  3c =  9
     3a +  5c + -3c =  4

If the file fails to open, print a descriptive error message stating what happened (include the name of the file in the message) and quit the program. If the value of N (the first integer read in) is not within the range from 1 to 10, print a descriptive error message stating what happened (include the name of the file in the message) and quit the program. If you program encounters any errors when reading the data file (such as non-integer values or not enough integer values), print a descriptive error message stating what happened (include the name of the file in the message) and quit the program. The following is a list of data files. These data files do not test all possible inputs, so you should create some data files of your own.

When reading in the data, you may find it a good idea to store the equations in a two dimensional array. Since there may be at most 10 equations with 11 values (the coefficients and result) in each an array of size 10x11 would be large enough to store any possible series of equations read into your program. If the value of N is not 10, then just use the first N rows and N+1 columns of this array.

Once your program has read in the data file, print out the series of equations as given in the data file. The first variable should be the letter a and the remaining variables should follow in order though the alphabet (the second variable being b, the third variable being c, etc.). Your output should align the variables in neat columns.

Now your program should prompt the user to run the program in step mode or in continious mode. Have the user enter in either an s or a c. If the user enters something else, print an error message and reprompt the user. Continue doing this until the user enters an s or a c. In step mode, your program should print out the series of equations at regular points though your program as the series of equations is solved before printing the solution (or a message stating no solution exists). In continious mode, your program should just print the solution (or a message stating no solution exists).

To begin solving the series of equations, we must first make sure that the coeffecients for each variable are not all zero. This is the case when a column in the array that stores the coefficients are not all zeros. If this occurs, then the series does not really have N variables. Instead it has N-1 variables (assuming only one column is all zeros). Note that it is fine if the results of all of the equations are zero. When this occurs, print out an descriptive error message stating what happened and which variable was unused (include the name of the file in the message) and quit the program.

Alternate Algorithm

I found a different algorithm to deal with the validation of the sequence of equation. You may use whichever you like. I think the newer algorithm will be easier to code.

Original Algorithm and Description
Now we must first make sure that the coefficients "along the main diagonal" are not zero. The "main diagonal" here refers to the first coefficient of the first equation, the second coefficient of the second equation, etc. If one of these coefficients is zero, we must exchange that equation with another equation so that there will not be any zeros only the "main diagonal". Let us assume that the i-th equation has a zero for its i-th coefficient. We can exchange the i-th equation with the j-th equation, if the i-th coefficient in the j-th equation is not zero and the j-th coefficient in the i-th equation is not zero. If this j-th equation does not exist, print out an descriptive error message stating what happened and which equation caused the problem (include the name of the file in the message) and quit the program. If the j-th equation does exist, exchange the i-th and j-th equations. If your program is running in step mode, print a message stating that equations i and j were exchanged (specify the proper values for i and j in this message) and redisplay the series of equations showing this change. If your program is running is continious mode, only print a message stating that equations i and j were exchanged (specify the proper values for i and j in this message) but DO NOT redisplay the series of equations showing this change. To help understand this, look at the following series of equations.
```
    3a + 0b +  1c + 1d = 7
    2a + 0b +  0c + 1d = 3
    0a + 3b + -1c + 1d = 4
    1a + 1b +  1c + 1d = 7
```
The second equation has a zero coefficient for its second variable. This equation can't be exchanged with the first equation, since the first equation also has a zero for its second coefficient (this exchange would not resolve the problem). This equation can't be exanged with the third equation, since the second equation has a zero for its third coefficient (this exchange would just move the problem to the "new" third equation). We could exchange the second equation with the fourth equation and continue with the program. Now note the following series of equations.
```
    3a + 0b +  1c = 6
    2a + 0b +  0c = 2
    0a + 3b + -1c = 3
```
The second equation has a zero coefficient for its second variable. This equation can't be exchanged with the first equation, since the first equation also has a zero for its second coefficient (this exchange would not resolve the problem). This equation can't be exanged with the third equation, since the second equation has a zero for its third coefficient (this exchange would just move the problem to the "new" third equation). In this case, your program cannot resolve this problem with a single exchange. So your program should print an error message and quit. You may note that putting the second equation first, the third equation second and the first equation third would allow the program to be solved but your program is not required to figure this out.
Once your program has checked for and resolved the above situations, you may begin to translate the series of equations into the form needed to solve the problem. It will take N steps to solve the problem, one for each variable in the equation. For the i-th step, the coefficent of the i-th variable in the i-th equation should be changed to 1 and the coefficents of the i-th variables in the other equations should be zeroed out. If your program is running in step mode, you should print out the translated series of equations at the end of every step. This means you should print out N series of equations showing the progress toward the solution. If your program is running is continious mode, you should NOT print out these series of equations.
There are two more error cases your program must be able to handle. This occurs when the i-th coefficient of the i-th equation becomes zero as a result of zeroing out another coefficent. If this happens, then you must check the value of the result of the i-th equation. If the result is non-zero, then the series of equations has no solution. If this occurs, print out a descriptive error message stating what happened and which equation caused the problem (include the name of the file in the message) and quit the program. If the result is zero, the the series of equation has multiple solutions. What occurs here is that two equations in the series were multiples of each other. If this occurs, print out a descriptive error message stating what happened and which equation caused the problem (include the name of the file in the message) and quit the program.
New Algorithm
This algorithm performs a simplier "pivot" operation before processing the i-th equation instead of checking and rearranging all N equations before any processing is done. This also removes all of the i-th coeffecient of the j-th equation stuff.
Before processing the i-th equation, if the i-th coefficient is zero, check the i-th coeffecients for all equations from i to N and find the equation with the largest absolute value for its i-th coefficient. If the i-th coefficient is not zero, proceed as normal.
If this coeffecient with the largest absolute value is zero, then we have an error. If this happens, then you must check the value of the result of the i-th equation. If the result is non-zero, then the series of equations has no solution. If this occurs, print out a descriptive error message stating what happened and which equation caused the problem (include the name of the file in the message) and quit the program. If the result is zero, the the series of equation has multiple solutions.
If the coefficient with the largest absolute value is not zero, exchange this equation with the i-th equation and then process the equations as with this new i-th equation. Check out the following example. This series has already processed varaible a.
```
    1a + 0b +  1/3c = 2
    0a + 0b + -2/3c = -2
    0a + 3b +   -1c = 3
```
Before processing variable b, compare the coefficients for variable b from equations 2 and 3 (in this case i is 2 and N is 3). Equation 3 has the largest absolute value, so we exchange equations 2 and 3 to get:
```
    1a + 0b +  1/3c = 2
    0a + 3b +   -1c = 3
    0a + 0b + -2/3c = -2
```
We can now proceed to set the 2nd coefficient of the 2nd equation to 1 and zero out the 2nd coefficients in the other equation.
When your program is running is step mode, it should print out the series of equations twice for each variable. Once after it has does o the exchange of equations and again after zeroing-out the other equations.

If your program makes it to the final form (the i-th coefficient of the i-th equation is 1 and the other coefficient are zero for all equations), the result values (in column N+1) are the solutions to the series of equations. This result value in the first equation is the value of the first variable, the result value in the second equation is the value for the second variable, etc. Your program is to print out these values is some readable form when your program is running in both step mode and continious mode. Once your program has print ou the solution, your program may quit.

Fraction Class

To earn full credit for this assignment, create a class called Fraction that will keep store all of the coefficients and results in the form of a fraction (an integer numerator and an integer denominator). This fraction must always express the number with the lowest possible denominator. When printing out the fraction, if the denominator is 1, only print the numerator. If the denominator is not 1, print the numerator followed immediately by the slash characher '/' followed immediately by the denomintor (no spaces before or after the slash). This Fraction class will be worth about 30% of the assignment. Those programs that solve the series of equations using floating point numbers will lose these points. However, it may be wise to first write a program that solves problem using an array of floating point numbers and then replace the floating point numbers with fractions after the rest of the program is working.

Your program is to be written using functions and methods. You should have a function for reading in the series of equations and another for printing the series of equations. Your program should have other functions, but you must decide what they should be. Perhaps you can have a function for changing a coefficient to 1 and another for zeroing out a coefficient. Other suggestions include functions for checking for the error cases involving coefficients with zero value.

Your program must be written using good programming style. This includes (but is not limitted to) such items as:

A header block comment that explains who wrote the program, why and what the program does.
A function block comment for each function that explains what the function does, what is returns and what each parameter is.
In-line comments that explain specific sections of code
Blank lines
Indentation
Meaningful identifer names
Readable and understandable output

Your program will be submitted electronically on the icarus system using the turnin command. The project name for this will be mp4. Your program will graded based on how it executes on the icarus system using the g++ compiler. Getting a program to work on another system or using another compiler does NOT mean it will work on the icarus system using the g++ compiler. Programs that do not work properly on the icarus system using the g++ compiler will lose points. Also programs that are not submitted via the turnin command will also lose points. Emailing a program is not an acceptable form of electronic submission.