Real Answer™ can use iteration to find values for the variables which make ALL the “=” equations in the Simultaneous Equation Set window true.
When you choose “Solve Eq. Set”. from the Solve menu, the simultaneous equation solver part of the Real Answer program begins an iterative search for a solution. Real Answer displays the variable values it tries and iteration number in the Solution Search window. If you are running out of memory, Real Answer saves memory by only showing the iteration number instead of maintaining a Solution Search window.
Once an iterative search for a solution has begun, you can abort the search at any time by holding down the “⌘” key while pressing the period (“.”) key.
You can adjust the criteria used by Real Answer to determine when to automatically stop searching for a solution to a set of simultaneous equations; choose the “Iterations” item from the “Solve” menu. An ensuing dialog box shows the termination conditions. If any one of those conditions is reached, Real Answer will terminate the iterative solution search. Iff the solution search terminates because the tolerance criterion is satisfied, the solution is considered to be valid, and the variables are assigned the values found by iterative solution. If the search is terminated without the tolerance criterion being satisfied, the search is considered unsuccessful and the variables that had their values perturbed by the iteration process are reset to their original values. (Note that some variables may have their values changed even if the solution search is unsuccessful; variables that are initialized to some starting value in the Simultaneous Equation Set window, and variables whose solution value can be calculated without iteration, all remain affected by a search for a solution even if it fails.)
Here is how the tolerance criterion is tested: every iteration, the values of the variables change. As Real Answer closes in on a solution, the variables change only a little bit from one iteration to the next as they approach their exact solution values. When all the variables change by less than the proportion specified by the tolerance field of the “Iterations” dialog box, the search process successfully terminates.
You can control the value a variable has at the first iteration. If a variable is initialized with an assignment statement (e.g. “x:=55”) anywhere in the Simultaneous Equation Set window, it will start the iteration process from that assigned value. Otherwise it will start from any value remaining in that variable from previous operations in Real Answer, or a preselected small value-- one millionth, if the variable has never been used before.
Many equations have more than one solution. By using assignment statements to initialize variables to values known to be close to a solution, you can cause Real Answer to find that particular solution.
The Simultaneous Equation Set window can be used to find roots of single equations such as polynomials. For example, typing (or copying) this into the Simultaneous Equation Set window:
50=x^4-3*x^3+2*x*x+6
x:=2
and then choosing “Solve Eq. Set”. from the Solve menu will cause Real Answer to find a solution.
The Simultaneous Equation Set window can also be used to solve groups of equations that do not need iteration. Typically the first equation will determine the value of one variable, the next equation will use that previously determined variable to set the value of another variable, and so on. Here’s an example:
