fsolve - mathtamuedu

fsolve - Penn Math

The syntax of fsolve is the standard Maple syntax: where "what" stands for the equation (or system of equations) to be solved and "how" refers to the variable(s) being solved for

fsolve - solve using floating-point arithmetic Calling Sequence: fsolve( eqns, vars, options ); Parameters: eqns - an equation or set of equations, or a procedure vars - (optional) an unknown or set of unknowns

Newton’s Method and fsolve

Maple Lab for Calculus I Week 11 Newton’s Method and fsolve Douglas B Meade Department of Mathematics Overview The analysis of a function via calculus involves solving a variety of equations: f0(x) = 0

Solve, RootOf, fsolve, isolve

Solve, RootOf, fsolve, isolve Maple is capable of solving a huge class of equations: (the solution tells us that can be arbitrary) One may extract the solutions using the "[ ]" notation (we will learn more about "[ ]" in the programming part of the lecture): (Note the use of ";" to separate statements We will be seeing that again when we discuss

Section 4: Solving Equations

Example 1) For other equations fsolve can be used to get one solution at a time (see Examples 2 and 3) Example 1 Maple's fsolve command will compute a numerical approximation for each solution of a polynomial equation To obtain approximate solutions for the equation x4Kx3K17 x2K6 xC2 =0, that is, for eqn := x^4-x^3-17*x^2-6*x+2=0; use fsolve

Critical Points, fsolve, and Custom Functions

Maple Lab for Calculus I Lab K Critical Points, fsolve, and Custom Functions Douglas Meade and Ronda Sanders Department of Mathematics Overview The analysis of a function via calculus involves solving a variety of equations: f0(x)=0 for critical numbers, f00(x) = 0 for possible in°ection points These equations are generally

Maple Lab, Week 18 Newton’s Method and Maple Programming

a command for solving equations, fsolve? Of course, the easy answer is that you may not always have Maple available to you Another answer is that Newton’s method might be faster than fsolve, and if so, this could be signi cant if you have a lot of equations to solve This exercise will be to use Maple’s timecommand to nd out which is faster,

Section 5: Functions: Defining, Evaluating and Graphing

d) Use Maple's fsolve command to approximate all solutions to the equation k(x) =4 Student Workspace 5 7 Answer 5 7 a) First declare the function via k := x -> x+3*sin(2*x); then plot using the plot command, obtaining plot(k(x),x=-1 8); b) As it can be seen from the following, there appears to be three intersection points at x = 3 25 ,4 825

Lesson 10: Polynomials

As we've seen, we can ask Maple to solve this system of equations for the two variables x and y We could try either fsolve or solve On a system of equations, fsolve will only return one solution (it's just for a single polynomial that it would return all the real solutions) fsolve({p1=0,p2=0},{x,y});

Math 2310 - Applied Differential Equations I Lab Session 3

Finally, use the "fsolve" routine available in Maple to obtain a closer approximation to the endpoints on the interval of validity From the qualitative results, a first approximation of the interval of validity is (-1 45, 4 65) The following two calls provide more accurate approximations of those endpoints > fsolve(13-4*exp(x)+4*x^3=0, x,-2

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fsolve Use fsolve to have Maple use numerical approximation techniques (as opposed to algebraic methods) to find a decimal approximation to a solution of an equation or system of equations. The syntax of fsolve is the standard Maple syntax: fsolve(what,how); where "what" stands for the equation (or system of equations) to be solved and "how" refers to the variable(s) being solved for. To read more about setting up equations for solution, see the description of the solve command. Using fsolve to solve a single equation: Because fsolve uses numerical techniques rather than algebraic ones, it is required that the number of equations be precisely the same as the number of variables being solved for. So, when fsolving one equation, it is crucial that there be exactly one unspecified variable in the equation. There are two ways to use the fsolve command. The first is precisely like the solve command: > fsolve(x^2+5*x=17,x); ,-7.3218253802.321825380 This demonstrates that fsolve, like solve, knows how many roots to expect of a polynomial and will attempt to find them all (even if some are complex). When solving a transcendental equation, fsolve is usually content to find one solution. The second way to use fsolve is especially important when an equation has many solutions, and you want to pick out a specific one. In this version of fsolve, it is possible to specify a domain (interval) in which the solution should be found. For example, consider the equation > eqn:=x=tan(x); := eqn = x()tanx As we can see from plotting both sides, this equation has many solutions: > plot({x,tan(x)},x=-8..8,-8..8);x8642-2-4-6-88 6 4 2 0 -2 -4 -6 -8 Just using fsolve on this equation will find one solution: > fsolve(eqn,x);

4.493409458

Now, suppose we want to find the solution between 6 and 8. Then we enter: > fsolve(eqn,x,x=6..8);

7.725251837

> tan(");

7.725251841

So we have found the solution to about 7 decimal places. What can go wrong? Aside from syntax errors, there are a few things that can go wrong when using fsolve. First, as with solve, it is possible that the "variable" in the equation to be solved has already been given a value (perhaps one that was forgotten in the course of the Maple session). This results in the following response: > x:=3: fsolve(x^2=4,x);

Error, (in fsolve) invalid arguments

The other things that can go wrong involve Maple's seeming inability to find a solution. This can result from one of two situations: first, there might be no solution -- second, the numerical procedure being used by Maple might need a little assistance from the user. For example: > x:='x':fsolve(sin(x)=exp(x^2),x); ()fsolve, = ()sinxe()x2 x This "non-response" from Maple indicates that it cannot find a solution. But that is because this equation has no solutions. Sometimes, fsolve chooses its initial approximation poorly and subsequently is unable to find a solution even if it exists. In this case Maple returns a message to this effect, and suggests choosing a "different" starting interval. In this case, using the second version of fsolve with specified domain will remedy the problem. (Of course, to find the appropriate domain, the most reasonable thing to do is plot the two sides of the equation and look for the intersection point!). Finally , fsolve will return an error message if there is a different number of equations than unknowns: > fsolve(a*x=1,x); Error, (in fsolve) should use exactly all the indeterminates Of course, the solve command is able to handle this equation easily. > solve(a*x=1,x);1 a Using fsolve to solve systems of equations: To be consistent with its "what","how" syntax, fsolve requires that a system of equations be enclosed in braces { } and that the list of variables being solved for also be so enclosed. For example: > fsolve({2*x+y=17,x^2-y^2=20},{x,y}); {}, = y-15.75516397 = x16.37758198 It is important to remember that the number of equations must be the same as the number of unknowns, and that no other (unspecified) variables are allowed in the equations: > fsolve({a*x+y=13,b*x-y=20},{x,y}); Error, (in fsolve) should use exactly all the indeterminates Finally, we note that it is possible (and often advisable) to give fsolve a domain to search in -- this is done by giving an interval for each variable separately, thereby providing fsolve with a rectangular box-like region in which to find a solution. For example, let's consider the system solved above: > eqn1:=2*x+y=17: eqn2:=x^2-y^2=20: We give the equations names mostly to remind you that this is possible. Experimenting around with several plots ultimately resulted in the following one, which shows that our "fsolve" statement above returned only one of the two intersection points: > plot({17-2*x,sqrt(x^2-20),-sqrt(x^2-20)},x=4..20,color=blue);x20181614121086420 10 0 -10 -20 We can force fsolve to find the leftmost one as follows: > fsolve({eqn1,eqn2},{x,y},{x=4..8,y=0..10}); {}, = y4.421830634 = x6.289084683 The syntax here is important! First comes the set of equations to solve (enclosed in braces), then the set of variables to solve for (enclosed in braces) and then the list of ranges for the variables (enclosed in braces). Only the third of these (the list of variable ranges) is optional when solving systems of equations. The other two must be present.

As usual with

solve and fsolve, we can substitute this solution (just as it is!) back into the equations to make sure it is correct. First, we give it a name: > s:="; := s{}, = y4.421830634 = x6.289084683 > subs(s,eqn1),subs(s,eqn2); , = 17.000000017 = 19.999999920

So it seems to work.