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14 Pages

### solve

Course: CSC 242, Fall 2009
School: Virgin Islands
Rating:

Word Count: 607

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restart: # &gt; This worksheet describes using solve and fsolve to solve equations. # Consider the solution of a single equation for a single variable. &gt; solve(sin(x)+y^2-2,x); / 2 \ -arcsin\y - 2/ # Solve for y; you get a sequence of 2 solutions &gt; sy:=solve(sin(x)+y^2-2,y); (1/2) (1/2)...

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restart: # > This worksheet describes using solve and fsolve to solve equations. # Consider the solution of a single equation for a single variable. > solve(sin(x)+y^2-2,x); / 2 \ -arcsin\y - 2/ # Solve for y; you get a sequence of 2 solutions > sy:=solve(sin(x)+y^2-2,y); (1/2) (1/2) sy := (2 - sin(x)) , -(2 - sin(x)) # Refer to the 2nd solution in the sequence using [2] > sy[2]; (1/2) -(2 - sin(x)) # Note that the solution implicitly assumes the expression for the # equation is set equal to zero. If we add a second independent # equation, we have > solve({sin(x)+y^2-2,y-ln(x)},{x,y}); # We don't get a solution, because there is no analytic answer to this # set of coupled equations; we can only determine a numerical answer, # e.g. using fsolve. > fsolve({sin(x)+y^2-2,y-ln(x)},{x,y}); {x = 0.2678513736, y = -1.317323028} # Is this the only solution? It would be interesting to plot the # functions to determine if that is the case. To do this we need the # plots package. > with(plots): # Warning, the name changecoords has been redefined # The plot command takes an equation as its first argument, the # variable range as its second (and optional third), and additional # plot options as the final argument. > implicitplot([sin(x)+y^2-2,y-ln(x)],x=0..10,y=-3..5,color=[red,blue],n > umpoints=2500); # So it looks like there's another solution around x=5; let's try to # find it. > fsolve({sin(x)+y^2-2,y-ln(x)},{x,y},x=4..6); {x = 5.342510434, y = 1.675695661} > fsolve({sin(x)+y^2-2,y-ln(x)},{x,y},x=0..1); {x = 0.2678513736, y = -1.317323028} # You can also predefine the equations used by solve or fsolve. > f:=sin(x)+y^2-2: > g:=y-ln(x); g := y - ln(x) > fsolve({f,g},{x,y},x=4..6); {y = 1.675695661, x = 5.342510434} # Notice that the order of x and y in the return set is different than # above; this underlines the fact that order is not preserved in sets. # The above can also use the functional operator specification of # functions. For example > x:='x':y:='y': > f:=(x,y)->sin(x)+y^2-2: > g:=(x,y)->y-ln(x); g := (x, y) -> y - ln(x) > fsolve({f(w,z),g(w,z)},{w,z},w=4..6); {w = 5.342510434, z = 1.675695661} # You cannot, however, omit the arguments of f and g: > fsolve({f,g},{x,y},x=4..6); # Error, (in fsolve) Case of systems of equations involving procedures # is not implemented yet # On the other hand, you can use a richer set of arguments > fsolve({f(x^2,y^2),g(x^2,y^2)},{x,y},x=2..3,y=1..2); {x = 2.311387123, y = 1.294486640} > fsolve({f(x^2,y^2),g(x^2,y^2)},{x,y},x=2..3,y=-2..1); {y = -1.294486640, x = 2.311387123} # Switching to a polynomial example, we encounter RootOf and allvalues: > f:=x->3*x^5-4*x-1; 5 f := x -> 3 x - 4 x - 1 > solve(f(x)); / 4 3 2 \ -1, RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 1/, / 4 3 2 \ RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 2/, / 4 3 2 \ RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 3/, / 4 3 2 \ RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 4/ > RootOf(f(x)); / 5 \ RootOf\3 _Z - 4 _Z - 1/ > allvalues(%); / | 1 1 (1/2) | -1, - + -- 3 | 4 12 | | \ (1/3) (2/3) \ // / (1/2)\ / (1/2)\ | || -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| 1 || -------------------------------------------------------------|^(1/2) + -- || (1/3) | 12 || / (1/2)\ | || \140 + 4 2089 / / \\ / (1/3) | / (1/2)\ | -30 \140 + 4 2089 / | | | \ (1/3) (2/3) \ / / (1/2)\ / (1/2)\ | | -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| | -------------------------------------------------------------|^(1/2) - 6 | (1/3) | | / (1/2)\ | | \140 + 4 2089 / / \ (1/3) (2/3) \ / (1/2)\ / (1/2)\ | -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| -------------------------------------------------------------|^(1/2) (1/3) | / (1/2)\ | \140 + 4 2089 / / (2/3) / (1/2)\ \140 + 4 2089 / / (1/3) (2/3) \ | / (1/2)\ / (1/2)\ | |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| + 144 |-------------------------------------------------------------|^(1/2) | (1/3) | | / (1/2)\ | \ \140 + 4 2089 / / \// / (1/3)| | (1/3) | (1/2) / (1/2)\ | |/ (1/2)\ | + 90 3 \140 + 4 2089 / | |\140 + 4 2089 / | | | | | | | / \ \ (1/3) (2/3) \ \\ / (1/2)\ / (1/2)\ | || -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| || -------------------------------------------------------------|^(1/2)||^(1/2), (1/3) | || / (1/2)\ | || \140 + 4 2089 / / // / (1/3) (2/3) | / (1/2)\ / (1/2)\ 1 1 (1/2) |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48 - + -- 3 |------------------------------------------------------------- 4 12 | (1/3) | / (1/2)\ \ \140 + 4 2089 / \ // | || | 1 || |^(1/2) - -- || | 12 || | || / \\ / (1/3) | / (1/2)\ | -30 \140 + 4 2089 / | | | \ (1/3) (2/3) \ / / (1/2)\ / (1/2)\ | | -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| | -------------------------------------------------------------|^(1/2) - 6 | (1/3) | | / (1/2)\ | | \140 + 4 2089 / / \ (1/3) (2/3) \ / (1/2)\ / (1/2)\ | -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| -------------------------------------------------------------|^(1/2) (1/3) | / (1/2)\ | \140 + 4 2089 / / (2/3) / (1/2)\ \140 + 4 2089 / / (1/3) (2/3) \ | / (1/2)\ / (1/2)\ | |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| + 144 |-------------------------------------------------------------|^(1/2) | (1/3) | | / (1/2)\ | \ \140 + 4 2089 / / \// / (1/3)| | (1/3) | (1/2) / (1/2)\ | |/ (1/2)\ | + 90 3 \140 + 4 2089 / | |\140 + 4 2089 / | | | | | | | / \ \ (1/3) (2/3) \ \\ / (1/2)\ / (1/2)\ | || -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| || -------------------------------------------------------------|^(1/2)||^(1/2), (1/3) | || / (1/2)\ | || \140 + 4 2089 / / // / (1/3) (2/3) | / (1/2)\ / (1/2)\ 1 1 (1/2) |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48 - - -- 3 |------------------------------------------------------------- 4 12 | (1/3) | / (1/2)\ \ \140 + 4 2089 / \ // / | || (1/3) | | 1 || / (1/2)\ | |^(1/2) + -- I ||30 \140 + 4 2089 / | | 12 || | | || | / \\ \ (1/3) (2/3) \ / / (1/2)\ / (1/2)\ | | -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| | -------------------------------------------------------------|^(1/2) + 6 | (1/3) | | / (1/2)\ | | \140 + 4 2089 / / \ (1/3) (2/3) \ / (1/2)\ / (1/2)\ | -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| -------------------------------------------------------------|^(1/2) (1/3) | / (1/2)\ | \140 + 4 2089 / / (2/3) / (1/2)\ \140 + 4 2089 / / (1/3) (2/3) \ | / (1/2)\ / (1/2)\ | |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| - 144 |-------------------------------------------------------------|^(1/2) | (1/3) | | / (1/2)\ | \ \140 + 4 2089 / / \// / (1/3)| | (1/3) | (1/2) / (1/2)\ | |/ (1/2)\ | + 90 3 \140 + 4 2089 / | |\140 + 4 2089 / | | | | | | | / \ \ (1/3) (2/3) \ \\ / (1/2)\ / (1/2)\ | || -5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| || -------------------------------------------------------------|^(1/2)||^(1/2), (1/3) | || / (1/2)\ | || \140 + 4 2089 / / // / ...

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