Solve symbolic equations numerically - MATLAB vpasolve (2024)

Open Live Script

syms xS = vpasolve(2*x^4 + 3*x^3 - 4*x^2 - 3*x + 2 == 0, x)

S =
$$\left(\begin{array}{c}-2.0\\ -1.0\\ 0.5\\ 1.0\end{array}\right)$$

S = vpasolve(sin(x) == 1/2, x)

S =$$0.52359877559829887307710723054658$$

Open Live Script

syms xeqnLeft = 200*sin(x);eqnRight = x^3 - 1;fplot([eqnLeft eqnRight])title([texlabel(eqnLeft) ' = ' texlabel(eqnRight)])

S1 = vpasolve(eqnLeft == eqnRight, x)

S1 =$$-0.0050000214585835715725440675982988$$

S2 = vpasolve(eqnLeft == eqnRight, x, -3)

S2 =$$-3.0009954677086430679926572924945$$

S3 = vpasolve(eqnLeft == eqnRight, x, 4)

S3 =$$3.0098746383859522384063444361906$$

Open Live Script

syms u vY = vpasolve([v^3 + 2*u == v, v^2 == u], [u,v])

Y = struct with fields: u: [3x1 sym] v: [3x1 sym]

uSol = Y.u

uSol =
$$\left(\begin{array}{c}0\\ 5.8284271247461900976033774484194\\ 0.1715728752538099023966225515806\end{array}\right)$$

vSol = Y.v

vSol =
$$\left(\begin{array}{c}0\\ -2.4142135623730950488016887242097\\ 0.4142135623730950488016887242097\end{array}\right)$$

syms xeqns = [3*x+2, 3*x+1];Y = vpasolve(eqns,x)

 Y = Empty sym: 0-by-1 

Open Live Script

syms x y[sol_x, sol_y] = vpasolve([x*sin(10*x) == y^3, y^2 == exp(-2*x/3)], [x,y])

sol_x =$$88.90707209659114864849280774681$$

sol_y =$$0.00000000000013470479710676694388973703681918$$

Open Live Script

syms xS = vpasolve(x^6 - x^2 == 3, x)

S =
$$\left(\begin{array}{c}-1.2929423350084724369196550436382\\ 1.2929423350084724369196550436382\\ -0.50188125716943915856832436499602-1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ -0.50188125716943915856832436499602+1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ 0.50188125716943915856832436499602-1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ 0.50188125716943915856832436499602+1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\end{array}\right)$$

assume(x,'real')S = vpasolve(x^6 - x^2 == 3, x)

S =
$$\left(\begin{array}{c}-1.2929423350084724369196550436382\\ 1.2929423350084724369196550436382\\ -0.50188125716943915856832436499602-1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ -0.50188125716943915856832436499602+1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ 0.50188125716943915856832436499602-1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ 0.50188125716943915856832436499602+1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\end{array}\right)$$

S = vpasolve(x^6 - x^2 == 3, x, [-Inf Inf])

S =
$$\left(\begin{array}{c}-1.2929423350084724369196550436382\\ 1.2929423350084724369196550436382\end{array}\right)$$

S = vpasolve(x^6 - x^2 == 3, x, [0 Inf])

S =$$1.2929423350084724369196550436382$$

S = vpasolve(x^6 - x^2 == 3, x, [-1 1+2i])

S =
$$\left(\begin{array}{c}-0.50188125716943915856832436499602+1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\\ 0.50188125716943915856832436499602+1.0429452224956770037495194222175\hspace{0.17em}\mathrm{i}\end{array}\right)$$

Open Live Script

syms x yeqn1 = exp(-x^2-y^2)*(x-4) - exp((-x^2-y^2)/2)*(x-2) == 0

eqn1 =
$${\mathrm{e}}^{-x^2-y^2}\hspace{0.17em}\left(x-4\right)-{\mathrm{e}}^{-\frac{x^2}{2}-\frac{y^2}{2}}\hspace{0.17em}\left(x-2\right)=0$$

eqn2 = exp(-x^2-y^2)*(y-2) - exp((-x^2-y^2)/2)*(y-4) == 0

eqn2 =
$${\mathrm{e}}^{-x^2-y^2}\hspace{0.17em}\left(y-2\right)-{\mathrm{e}}^{-\frac{x^2}{2}-\frac{y^2}{2}}\hspace{0.17em}\left(y-4\right)=0$$

[solX, solY] = vpasolve([eqn1 eqn2],[x y])

 solX = Empty sym: 0-by-1 solY = Empty sym: 0-by-1 

[solX, solY] = vpasolve([eqn1 eqn2],[x y],[2; 4])

solX =$$1.9999092125057125429174334656647$$

solY =$$4.0000907874942874570825665343353$$

Open Live Script

syms xf = x-tan(x);for n = 1:3 S = vpasolve(f,x)end

S =$$0$$

S =$$0$$

S =$$0$$

for n = 1:3 S = vpasolve(f,x,'Random',true)end

S =$$-227.76107684764829218924973598808$$

S =$$102.09196646490764333652956578441$$

S =$$61.244730260374400372753016364097$$

S = vpasolve(f,x,[10 12],'Random',true)

S =$$10.904121659428899827148702790189$$

• If init_param is a number or, in the case of multivariate equations, a vector of numbers, then the numeric solver uses it as an initial guess. If init_param is specified as a scalar while the system of equations is multivariate, then the numeric solver uses the scalar value as an initial guess for all variables. For an example, see Find Multiple Solutions by Specifying Initial Guesses.

• If init_param is a matrix with two columns, then the two entries of the rows specify the bounds of an initial guess for the corresponding variables. To specify an initial guess in a matrix of search ranges, specify both columns as the initial guess value.

• If you specify init_param as a search range [a b] and the values a,b are complex numbers, then vpasolve searches for the solutions in the rectangular search area in the complex plane. Here, a specifies the bottom-left corner of the rectangular search area, and b specifies the top-right corner of that area. For an example, see Specify Ranges of Solutions.

• To omit a search range for a variable, set the search range for that variable to [NaN, NaN] in init_param. All other uses of NaN in init_param will error.