Documentation

Solve Equations Numerically

The Symbolic Math Toolbox™ offers both numeric and symbolic equation solvers. For a comparison of numeric and symbolic solvers, please see Select Numeric or Symbolic Solver. An equation or a system of equations can have multiple solutions. To find these solutions numerically, use the function vpasolve. For polynomial equations, vpasolve returns all solutions. For nonpolynomial equations, vpasolve returns the first solution it finds. This shows you how to use vpasolve to find solutions to both polynomial and nonpolynomial equations, and how to obtain these solutions to arbitrary precision.

Find All Roots of a Polynomial Function

Use vpasolve to find all the solutions to function $f\left(x\right)=6{x}^{7}-2{x}^{6}+3{x}^{3}-8$.

syms f(x)
f(x) = 6*x^7-2*x^6+3*x^3-8;
sol = vpasolve(f)
sol =
1.0240240759053702941448316563337
- 0.88080620051762149639205672298326 + 0.50434058840127584376331806592405i
- 0.88080620051762149639205672298326 - 0.50434058840127584376331806592405i
- 0.22974795226118163963098570610724 + 0.96774615576744031073999010695171i
- 0.22974795226118163963098570610724 - 0.96774615576744031073999010695171i
0.7652087814927846556172932675903 + 0.83187331431049713218367239317121i
0.7652087814927846556172932675903 - 0.83187331431049713218367239317121i

vpasolve returns seven roots of the function, as expected, because the function is a polynomial of degree seven.

Find Zeros of a Nonpolynomial Function Using Search Ranges and Starting Points

Consider the function $f\left(x\right)={e}^{\left(x/7\right)}\mathrm{cos}\left(2x\right)$. A plot of the function reveals periodic zeros, with increasing slopes at the zero points as x increases.

syms x
h = fplot(exp(x/7)*cos(2*x),[-2 25]);
grid on Use vpasolve to find a zero of the function f. Note that vpasolve returns only one solution of a nonpolynomial equation, even if multiple solutions exist. On repeated calls, vpasolve returns the same result, even if multiple zeros exist.

f = exp(-x/20)*cos(2*x);
for i = 1:3
vpasolve(f,x)
end
ans =
19.634954084936207740391521145497
ans =
19.634954084936207740391521145497
ans =
19.634954084936207740391521145497

To find multiple solutions, set the option random to true. This makes vpasolve choose starting points randomly. For information on the algorithm that chooses random starting points, see Algorithms on the vpasolve page.

for i = 1:3
vpasolve(f,x,'random',true)
end
ans =
-226.98006922186256147892598444194
ans =
98.174770424681038701957605727484
ans =
58.904862254808623221174563436491

To find a zero close to x = 10 and to x = 1000, set the starting point to 10, and then to 1000.

vpasolve(f,x,10)
vpasolve(f,x,1000)
ans =
10.210176124166828025003590995658
ans =
999.8118620049516981407362567287

To find a zero in the range $15\le x\le 25$, set the search range to [15 25].

vpasolve(f,x,[15 25])
ans =
21.205750411731104359622842837137

To find multiple zeros in the range [15 25], you cannot call vpasolve repeatedly as it returns the same result on each call, as previously shown. Instead, set random to true in conjunction with the search range.

for i = 1:3
vpasolve(f,x,[15 25],'random',true)
end
ans =
21.205750411731104359622842837137
ans =
16.493361431346414501928877762217
ans =
16.493361431346414501928877762217

If you specify the random option while also specifying a starting point, vpasolve warns you that the two options are incompatible.

vpasolve(f,x,15,'random',true)
Warning: 'Random' has no effect because
all variables have a starting value.
> In sym/vpasolve (line 168)
ans =
14.922565104551517882697556070578

Create the function findzeros below to systematically find all zeros for f in a given search range, within the error tolerance. It starts with the input search range and calls vpasolve to find a zero. Then, it splits the search range into two around the zero’s value, and recursively calls itself with the new search ranges as inputs to find more zeros. The first input is the function, the second input is the range, and the optional third input allows you to specify the error between a zero and the higher and lower bounds generated from it.

The function is explained section by section here.

Declare the function with the two inputs and one output.

function sol = findzeros(f,range,err)

If you do not specify the optional argument for error tolerance, findzeros sets err to 0.001.

if nargin < 2
err = 1e-3;
end

Find a zero in the search range using vpasolve.

sol = vpasolve(f,range);

If vpasolve does not find a zero, exit.

if(isempty(sol))
return

If vpasolve finds a zero, split the search range into two search ranges above and below the zero.

else
lowLimit = sol-err;
highLimit = sol+err;

Call findzeros with the lower search range. If findzeros returns zeros, copy the values into the solution array and sort them.

temp = findzeros(f,[range(1) lowLimit],1);
if ~isempty(temp)
sol = sort([sol temp]);
end

Call findzeros with the higher search range. If findzeros returns zeros, copy the values into the solution array and sort them.

temp = findzeros(f,[highLimit range(2)],1);
if ~isempty(temp)
sol = sort([sol temp]);
end
return
end
end

The entire function findzeros is as follows.

function sol = findzeros(f,range,err)
if nargin < 3
err = 1e-3;
end
sol = vpasolve(f,range);
if(isempty(sol))
return
else
lowLimit = sol-err;
highLimit = sol+err;
temp = findzeros(f,[range(1) lowLimit],1);
if ~isempty(temp)
sol = sort([sol temp]);
end
temp = findzeros(f,[highLimit range(2)],1);
if ~isempty(temp)
sol = sort([sol temp]);
end
return
end
end

Call findzeros with search range [10 20] to find all zeros in that range for f(x) = exp(-x/20)*cos(2*x), within the default error tolerance.

syms f(x)
f(x) = exp(-x/20)*cos(2*x);
findzeros(f,[10 20])
ans =
[ 10.210176124166828025003590995658, 11.780972450961724644234912687298,...
13.351768777756621263466234378938, 14.922565104551517882697556070578,...
16.493361431346414501928877762217, 18.064157758141311121160199453857,...
19.634954084936207740391521145497]

Obtain Solutions to Arbitrary Precision

Use digits to set the precision of the solutions. By default, vpasolve returns solutions to a precision of 32 significant figures. Use digits to increase the precision to 64 significant figures. When modifying digits, ensure that you save its current value so that you can restore it.

f = exp(x/7)*cos(2*x);
vpasolve(f)
digitsOld = digits;
digits(64)
vpasolve(f)
digits(digitsOld)
ans =
-7.0685834705770347865409476123789
ans =
-7.068583470577034786540947612378881489443631148593988097193625333

Solve Multivariate Equations Using Search Ranges

Consider the following system of equations.

$\begin{array}{l}z=10\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)\\ z=x+{y}^{2}-0.1{x}^{2}y\\ x+y-2.7=0\end{array}$

A plot of the equations for 0 ≤ x ≤ 2.5 and 0 ≤ x ≤ 2.5 shows that the three surfaces intersect in two points. To better visualize the plot, use view. To scale the colormap values, use caxis.

syms x y z
eqn1 = z == 10*(cos(x) + cos(y));
eqn2 = z == x+y^2-0.1*x^2*y;
eqn3 = x+y-2.7 == 0;
equations = [eqn1 eqn2 eqn3];
fimplicit3(equations)
axis([0 2.5 0 2.5 -20 10])
title('System of Multivariate Equations')
view(69, 28)
caxis([-15 10]) Use vpasolve to find a point where the surfaces intersect. The function vpasolve returns a structure. To access the solution, index into the structure.

sol = vpasolve(equations);
[sol.x sol.y sol.z]
ans = $\left(\begin{array}{ccc}2.3697477224547979209101337160174& 0.33025227754520207908986628398261& 2.2933543768232277431243854708612\end{array}\right)$

To search a region of the solution space, specify search ranges for the variables. If you specify the ranges $0\le x\le 1.5$ and $1.5\le y\le 2.5$, then vpasolve function searches the bounded area shown in the picture. Use vpasolve to find a solution for this search range$0\le x\le 1.5$ and $1.5\le y\le 2.5$. To omit a search range for z, set the search range to [NaN NaN].

vars = [x y z];
range = [0 1.5; 1.5 2.5; NaN NaN];
sol = vpasolve(equations, vars, range);
[sol.x sol.y sol.z]
ans = $\left(\begin{array}{ccc}0.91062661725633361176950031551069& 1.7893733827436663882304996844893& 3.9641015721356254724107884666807\end{array}\right)$

To find multiple solutions, you can set the random option to true. This makes vpasolve use random starting points on successive runs. The random option can be used in conjunction with search ranges to make vpasolve use random starting points within a search range. Because random selects starting points randomly, the same solution might be found on successive calls. Call vpasolve repeatedly to ensure you find both solutions.

clear sol
range = [0 3; 0 3; NaN NaN];
for i = 1:5
temp = vpasolve(equations, vars, range, 'random', true);
sol(i,1) = temp.x;
sol(i,2) = temp.y;
sol(i,3) = temp.z;
end
sol
sol =

$\left(\begin{array}{ccc}0.91062661725633361176950031551069& 1.7893733827436663882304996844893& 3.9641015721356254724107884666807\\ 2.3697477224547979209101337160174& 0.33025227754520207908986628398261& 2.2933543768232277431243854708612\\ 0.91062661725633361176950031551069& 1.7893733827436663882304996844893& 3.9641015721356254724107884666807\\ 0.91062661725633361176950031551069& 1.7893733827436663882304996844893& 3.9641015721356254724107884666807\\ 0.91062661725633361176950031551069& 1.7893733827436663882304996844893& 3.9641015721356254724107884666807\end{array}\right)$

Plot the equations. Superimpose the solutions as a scatter plot of points with yellow X markers using scatter3. To better visualize the plot, make two of the surfaces transparent using alpha. Scale the colormap to the plot values using caxis, and change the perspective using view.

vpasolve finds solutions at the intersection of the surfaces formed by the equations as shown.

clf
ax = axes;
h = fimplicit3(equations);
h(2).FaceAlpha = 0;
h(3).FaceAlpha = 0;
axis([0 2.5 0 2.5 -20 10])
hold on
scatter3(sol(:,1),sol(:,2),sol(:,3),600,'yellow','X','LineWidth',2)
title('Randomly found solutions in specified search range')
cz = ax.Children;
caxis([0 20])
view(69,28)
hold off Mathematical Modeling with Symbolic Math Toolbox

Get examples and videos