Maple/Simultaneous Equations

Introduction

This page focuses on using Maple to find both the symbolic and the numeric solutions to equations obtained from electric circuits. It assumes that you have already taken the steps in Maple/Initialization and Documentation to start Maple and begin documenting your work.

Very Basic Example

The example code below assumes you are running a worksheet in Maple. There is a finished example at the end of this section which also includes the commands to display different variables. Imagine you have the equation $$ax=d$$ and you want to solve for x. You can do this in Maple as follows:

Initialization

You are not explicitly required to include the restart command in a worksheet, but it does help if you end up making edits later and need to re-run everything from scratch. Go ahead and put

restart

as the first executable in your worksheet.

Define Equations

In Maple, the way you define a variable is by typing the name of the variable, followed by the symbols :=, followed by whatever items you want to store in the variable. Note the importance of the colon directly in front of the equals sign - without it, Maple will not assign a value to a variable but will merely print out the equation you typed. One benefit of this is you can define variables to hold on to equations and then use those variables later, in concert with Maple's solver, to get answers for the unknowns. Given that, we will define a variable eqn1 to store the equation $$ax=d$$:

eqn1:=a*x=d

Solve Equations

The easiest way to solve an equation (or a system) of equations is to use the solve command. The most formal, and flexible, way to use this command is to give it a set of equations (surrounded by curly brackets) and a list of variables (surrounded by square brackets). The result will be an expression, set of expressions, list of expressions, or list of list of expressions depending on the nature and number of the equations and the solutions. For example, if you add the code:

soln1 := solve({eqn1}, [x])

then Maple will produce a variable called soln1 that has a list with a list with an expression; specifically, $$soln1 := [[x = \frac{d}{a}]]$$

Make Substitutions

Now that you have symbolic answers, you can make numerical substitutions for those symbols using the subs command. The subs command expects a series of equalities to define the substitutions followed by a single item into which to make those substitutions. For example, to see what x is when d is 10, you can write:

subs(d = 10, soln1)

and you will get the new list of lists $$[[x = \frac{10}{a}]]$$. If you want to see multiple substitutions, you can put them all at the start of the soln command:

subs(a=3, d = 10, soln1)

will give $$[[x = \frac{10}{3}]]$$

More Complicated Example

Let us now assume that we want to solve the following equations:

\( \begin{align} ax+by+cz&=j\\ dx+ey+fz&=k\\ gx+hy+iz&=l \end{align} \)

where x, y, and z are unknowns, a through i are known coefficients, and j through l are known variables.

Initialization

Go ahead and put

restart

as the first executable in your worksheet.

Define Equations

Now there are three different equations to store; you could store all three in a single variable containing a set or you could define three variables and then include them in a set in a solve command. We will do the latter, so at the prompt, type:

eqn1:=a*x+b*y+c*z=j
eqn2:=d*x+e*y+f*z=k
eqn3:=g*x+h*y+i*z=l

Note that each time you hit return to go to the next line, Maple processes your input and reports back what it has done. It will also number the outputs for you so you can refer to them later. As an aside, if you try to copy and paste this whole block, when you hit return, you will get a parsing error. Maple is trying to interpret the whole group as one long string of code and gets confused at the end of the first equation. To fix this, if you copy and paste multiple lines of codes, you can end them with a semi-colon:

eqn1:=a*x+b*y+c*z=j;
eqn2:=d*x+e*y+f*z=k;
eqn3:=g*x+h*y+i*z=l;

The three lines will now be in a single execution group and all three will run when you hit return at the end of that line.

Whichever way you entered and ran the code, at this point, Maple now has three variables, each of which defined as an equation. It is perfectly happy having undefined items in the equations. Note that in Maple the complex number $$\sqrt{-1}$$ is given with a capital $$I$$; you must avoid using $$I$$ as a variable in Maple. We are using lower case $$i$$, which is fine.

Solve Equations With Maple

Now we just need to give the solve command a set of equations and a list of variables. Note that you need to give a complete list of the unknowns for a system even if yuo are only looking for the value of one of them. That is to say, even if we are just trying to solve $$x$$, we still need to let Maple know that $$x$$, $$y$$, and $$z$$ are the unknowns -- anything not listed as an unknown variable is considered known, and thus a system can become overconstrained and not have a solution. Add the line:

soln1:=solve({eqn1, eqn2, eqn3}, [x, y, z])

Hit return, and you will once again note that Maple produces a list of lists of expressions for the solution.

As an aside, if we had not included the variable list and instead had asked

soln2:=solve({eqn1, eqn2, eqn3})

Maple would have given all possible combinations of all 15 symbols that would satisfy the equations. Conversely, if we had given Maple only x to work with as an unknown by typing:

soln3:=solve({eqn1, eqn2, eqn3}, [x])

the answer would come back as an empty list empty because no value of x satisfies the three equations for arbitrary values of the other 14 variables.

Make Substitutions

Now that you have the symbolic answers to the variables x, y, and z, you may want to substitute the actual coefficient values to obtain a numerical solution. In the simple example above we put the substitution expressions directly into the subscommand. Given that we may want to re-use those expressions, or that we may have multiple sets of substitutions, another way to use the subs command is to generate a list of the known values then tell Maple to substitute in the numerical values. Add the following lines of code:

Vals := a=-1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, i=9, j=10, k=11, l=12
subs(Vals, soln1)

Remember if you cut and paste these into a single group you will need to add a semi-colon after the first line or else you will get a parsing error. The list of lists in soln1 will now be shown with numerical valuesinstead of symbols. Note that you have not made any actual changes to any of the variables - you have just asked Maple to show you what they would look like given the particular substitutions presented in Vals. This is a very powerful tool, since you can substitute in a variety of values to see how one or more parameters influence a particular variable or variables.

Multiple and Dependent Substitution Lists

If you have several sets of equations you want to use for substitution - including some which are dependent on values set in other equations, you can still use subs -- you just need to be careful about the order of the substitutions. As an example, imagine some variable:

\( \begin{align} m&=p+q \end{align} \)

where

\( \begin{align} p&=r*s\\ q&=t-u \end{align} \)

and

\( \begin{align} r&=1 & t&=3\\ s&=2 & u&=4 \end{align} \)

To get m in terms of r, s, t, and u, you could write:

eqn1:=m=p+q;
sublist1:=p=r*s, q=t-u;
subs(sublist1, eqn1);

If you want to get m's numerical value, you must first get m in terms of r, s, t, and u, and then you can substitute in the numbers for those variables. Specifically:

eqn1:=m=p+q;
sublist1:=p=r*s, q=t-u;
sublist2:=r=1, s=2, t=3, u=4;
subs(sublist1, sublist2, eqn1);

This will show $$m=1$$; again, Maple has not made the variable $$m$$ equal to the number 1, it is just showing what things would be ilke if the substitutions were made.

Putting the equations in the wrong order will end up yielding an answer that is still in terms of r, s, t, and u. The reason is that subs only makes substitutions into the last entry in the argument list. subs(sublist2, sublist2, eqn1); will yield $$m=rs+t-u$$.

Extracting Expressions from Lists (or Lists of Lists)

Sometimes, you will need to take equations out of a set of brackets to use them. For example, assume that you are looking back at the worksheet where you solved three linear algebra equations for $$x$$, $$y$$, and $$z$$. Imagine you have some variables you want to calculate called alpha and beta, which has formulas of:

\( \begin{align} \alpha&=x+y+z \beta=y*z \end{align} \)

You can put in the solutions for x, y, and z to get alpha and beta in terms of those characters. What makes this a bit difficult is that MySoln is given as a single-row matrix and subs just wants the equations themselves. If you just type soln1 in the worksheet, you will get a list of lists. To extract only the expressions, you can write:

soln1[][]

and you will instead see a comma-separated collecting of expressions without any brackets. The default index for a list is everything - you could also write

soln1[1][]

here and that would grab the first inner list and then all the items from that inner list (Maple is 1-indexed, similar to MATLAB and different frmo Python).

In the worksheet where you solved for the three equations, go ahead and add the lines

aux1:=alpha=x+y+z
aux2:=beta=y*x
subs(soln1[1][], {aux1, aux2})

to the end of your worksheet. Note that since we want all the substitutions to go into two different items, those items are collected in a set with curly brackets; you can also collect them in a list if you would like. The rest of this command will be a set (or list) showing your auxiliary equations using the symbols solved for in soln1.

Now imagine you want the numerical values for $$\alpha$$ and $$\beta$$ based on the symbolic solutions in soln1 and the numerical values in vals. Again - the order is important - you need to first substitute in the equations for the variables higher in the dependency list, then give values to the known quantities, then substitute all that into whatever is in the final argument of subs. If you try subs(vals, soln1[1][], {aux1, aux2}) you will not see any changes - the vals list will not find any of $$a$$ through $$l$$ in the auxiliary equations because those substitutions have not happened yet. On the other hand, if you run subs(vals, soln1[1][], {aux1, aux2}) you will get numerical values for $$\alpha$$ and $$\beta$$.

Cleaning Things Up

Many times, Maple will produce an expression that is more complicated than it needs to be. To get what it considers to be the simplest form, use the simplify(expand( )) compound function. The expand will take the expression and represent it using as many simple terms as necessary while simplify will recombine them in the most compact form. Finally, to get a decimal value, use the evalf[N]( ) function, where N represents the number of decimal digits to use. For example,

simplify(expand(alpha))

will produce the most symbolically simplified version of \(\alpha\) while

evalf[8](subs(Vals, alpha))

will produce a floating point result for \(\alpha\). With practice, you will see how best to combine evalf, simplify, and expand to get the form of answer you want.

Memory Issues

A major issue to consider with Maple is its memory. At the end of the worksheet above, there are several variables that are defined, including x, y, and z. If you go back near the beginning, click in the line where eqn1 is defined, and hit return, you will notice that where x, y, and z were before, their symbolic solutions from much further down the worksheet are being used. This is why the restart command is so helpful - if you need to to run a worksheet again, it is best to always start from scratch. A shortcut for running an entire worksheet is the !!! button at the top of the window.