# Documentation

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# vpasolve

Numeric solver

## Syntax

• S = vpasolve(eqn)
example
• S = vpasolve(eqn,var)
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• S = vpasolve(eqn,var,init_guess)
example
• Y = vpasolve(eqns)
• Y = vpasolve(eqns,vars)
example
• Y = vpasolve(eqns,vars,init_guess)
• [y1,...,yN] = vpasolve(eqns)
• [y1,...,yN] = vpasolve(eqns,vars)
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• [y1,...,yN] = vpasolve(eqns,vars,init_guess)
• ___ = vpasolve(___,Name,Value)
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## Description

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S = vpasolve(eqn) numerically solves the equation eqn for the variable determined by symvar.

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S = vpasolve(eqn,var) numerically solves the equation eqn for the variable specified by var.

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S = vpasolve(eqn,var,init_guess) numerically solves the equation eqn for the variable specified by var using the starting point or search range specified in init_guess. If you do not specify var, vpasolve solves for variables determined by symvar.
Y = vpasolve(eqns) numerically solves the system of equations eqns for variables determined by symvar. This syntax returns Y as a structure array. You can access the solutions by indexing into the array.

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Y = vpasolve(eqns,vars) numerically solves the system of equations eqns for variables specified by vars. This syntax returns a structure array that contains the solutions. The fields in the structure array correspond to the variables specified by vars.
Y = vpasolve(eqns,vars,init_guess) numerically solves the system of equations eqns for the variables vars using the starting values or the search range init_guess.
[y1,...,yN] = vpasolve(eqns) numerically solves the system of equations eqns for variables determined by symvar. This syntax assigns the solutions to variables y1,...,yN.

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[y1,...,yN] = vpasolve(eqns,vars) numerically solves the system of equations eqns for the variables specified by vars.
[y1,...,yN] = vpasolve(eqns,vars,init_guess) numerically solves the system of equations eqns for the variables specified by vars using the starting values or the search range init_guess.

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___ = vpasolve(___,Name,Value) numerically solves the equation or system of equations for the variable or variables using additional options specified by one or more Name,Value pair arguments.

## Examples

### Solve Polynomial Equation

For polynomial equations, vpasolve returns all solutions:

syms x vpasolve(4*x^4 + 3*x^3 + 2*x^2 + x + 5 == 0, x)
ans = - 0.88011377126068169817875190457835 + 0.76331583387715452512978468102263i - 0.88011377126068169817875190457835 - 0.76331583387715452512978468102263i 0.50511377126068169817875190457835 + 0.81598965068946312853227067890656i 0.50511377126068169817875190457835 - 0.81598965068946312853227067890656i

### Solve Nonpolynomial Equation

For nonpolynomial equations, vpasolve returns the first solution that it finds:

syms x vpasolve(sin(x^2) == 1/2, x)
ans = -226.94447241941511682716953887638

### Assign Solutions to Structure Array

When solving a system of equations, use one output argument to return the solutions in the form of a structure array:

syms x y S = vpasolve([x^3 + 2*x == y, y^2 == x], [x, y])
S = struct with fields: x: [6×1 sym] y: [6×1 sym]

Display solutions by accessing the elements of the structure array S:

S.x
ans = 0.2365742942773341617614871521768 0 - 0.28124065338711968666197895499453 + 1.2348724236470142074859894531946i - 0.28124065338711968666197895499453 - 1.2348724236470142074859894531946i 0.16295350624845260578123537890613 - 1.6151544650555366917886585417926i 0.16295350624845260578123537890613 + 1.6151544650555366917886585417926i
S.y
ans = 0.48638903593454300001655725369801 0 0.70187356885586188630668751791218 + 0.87969719792982402287026727381769i 0.70187356885586188630668751791218 - 0.87969719792982402287026727381769i - 0.94506808682313338631496614476119 + 0.85451751443904587692179191887616i - 0.94506808682313338631496614476119 - 0.85451751443904587692179191887616i

### Assign Solutions to Variables When Solving System of Equations

When solving a system of equations, use multiple output arguments to assign the solutions directly to output variables. To ensure the correct order of the returned solutions, specify the variables explicitly. The order in which you specify the variables defines the order in which the solver returns the solutions.

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

### Find Multiple Solutions by Specifying Starting Points

Plot the two sides of the equation, and then use the plot to specify initial guesses for the solutions.

Plot the left and right sides of the equation .

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

This equation has three solutions. If you do not specify the initial guess (zero-approximation), vpasolve returns the first solution that it finds:

vpasolve(200*sin(x) == x^3 - 1, x)
ans = -0.0050000214585835715725440675982988

Find one of the other solutions by specifying the initial point that is close to that solution:

vpasolve(200*sin(x) == x^3 - 1, x, -4)
ans = -3.0009954677086430679926572924945
vpasolve(200*sin(x) == x^3 - 1, x, 3)
ans = 3.0098746383859522384063444361906

### Specify Ranges for Solutions

You can specify ranges for solutions of an equation. For example, if you want to restrict your search to only real solutions, you cannot use assumptions because vpasolve ignores assumptions. Instead, specify a search interval. For the following equation, if you do not specify ranges, the numeric solver returns all eight solutions of the equation:

syms x vpasolve(x^8 - x^2 == 3, x)
ans = -1.2052497163799060695888397264341 1.2052497163799060695888397264341 - 0.77061431370803029127495426747428 + 0.85915207603993818859321142757163i - 0.77061431370803029127495426747428 - 0.85915207603993818859321142757164i -1.0789046020338265308047436284205i 1.0789046020338265308047436284205i 0.77061431370803029127495426747428 + 0.85915207603993818859321142757164i 0.77061431370803029127495426747428 - 0.85915207603993818859321142757163i

Suppose you need only real solutions of this equation. You cannot use assumptions on variables because vpasolve ignores them.

assume(x, 'real') vpasolve(x^8 - x^2 == 3, x)
ans = -1.2052497163799060695888397264341 1.2052497163799060695888397264341 - 0.77061431370803029127495426747428 + 0.85915207603993818859321142757163i - 0.77061431370803029127495426747428 - 0.85915207603993818859321142757164i -1.0789046020338265308047436284205i 1.0789046020338265308047436284205i 0.77061431370803029127495426747428 + 0.85915207603993818859321142757164i 0.77061431370803029127495426747428 - 0.85915207603993818859321142757163i

Specify the search range to restrict the returned results to particular ranges. For example, to return only real solutions of this equation, specify the search interval as [-Inf Inf]:

vpasolve(x^8 - x^2 == 3, x, [-Inf Inf])
ans = -1.2052497163799060695888397264341 1.2052497163799060695888397264341

Return only nonnegative solutions:

vpasolve(x^8 - x^2 == 3, x, [0 Inf])
ans = 1.2052497163799060695888397264341

The search range can contain complex numbers. In this case, vpasolve uses a rectangular search area in the complex plane:

vpasolve(x^8 - x^2 == 3, x, [-1, 1 + i])
ans = - 0.77061431370803029127495426747428 + 0.85915207603993818859321142757164i 0.77061431370803029127495426747428 + 0.85915207603993818859321142757164i

### Find Multiple Solutions for Nonpolynomial Equation

By default, vpasolve returns the same solution on every call. To find more than one solution for nonpolynomial equations, set random to true. This makes vpasolve use a random starting value which can lead to different solutions on successive calls.

If random is not specified, vpasolve returns the same solution on every call.

syms x f = x-tan(x); for n = 1:3 vpasolve(f,x) end
ans = 0 ans = 0 ans = 0

When random is set to true, vpasolve returns a distinct solution on every call.

syms x f = x-tan(x); for n = 1:3 vpasolve(f,x,'random',true) end
ans = -227.76107684764829218924973598808 ans = 102.09196646490764333652956578441 ans = 61.244730260374400372753016364097

random can be used in conjunction with a search range:

vpasolve(f,x,[10 12],'random',true)
ans = 10.904121659428899827148702790189

## Input Arguments

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Equation to solve, specified as a symbolic equation or symbolic expression. A symbolic equation is defined by the relation operator ==. If eqn is a symbolic expression (without the right side), the solver assumes that the right side is 0, and solves the equation eqn == 0.

Variable to solve equation for, specified as a symbolic variable. If var is not specified, symvar determines the variables.

System of equations or expressions to be solve, specified as a symbolic vector, matrix, or N-D array of equations or expressions. These equations or expressions can also be separated by commas. If an equation is a symbolic expression (without the right side), the solver assumes that the right side of that equation is 0.

Variables to solve system of equations for, specified as a symbolic vector. These variables are specified as a vector or comma-separated list. If vars is not specified, symvar determines the variables.

Initial guess for a solution, specified as a numeric value, vector, or matrix with two columns.

If init_guess is a number or, in the case of multivariate equations, a vector of numbers, then the numeric solver uses it as a starting point. If init_guess is specified as a scalar while the system of equations is multivariate, then the numeric solver uses the scalar value as a starting point for all variables.

If init_guess is a matrix with two columns, then the two entries of the rows specify the bounds of a search range for the corresponding variables. To specify a starting point in a matrix of search ranges, specify both columns as the starting point value.

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

By default, vpasolve uses its own internal choices for starting points and search ranges.

### Name-Value Pair Arguments

Example: vpasolve(x^2 - 4 == 0,x,'random',true)

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Use a random starting point for finding solutions, specified as a comma-separated pair consisting of random and a value, which is either true or false. This is useful when you solve nonpolynomial equations where there is no general method to find all the solutions. If the value is false, vpasolve uses the same starting value on every call. Hence, multiple calls to vpasolve with the same inputs always find the same solution, even if several solutions exist. If the value is true, however, starting values for the internal search are chosen randomly in the search range. Hence, multiple calls to vpasolve with the same inputs might lead to different solutions. Note that if you specify starting points for all variables, setting random to true has no effect.

## Output Arguments

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Solutions of univariate equation, returned as symbolic value or symbolic array. The size of a symbolic array corresponds to the number of the solutions.

Solutions of system of equations, returned as a structure array. The number of fields in the structure array corresponds to the number of variables to be solved for.

Variables that are assigned solutions of system of equations, returned as an array of numeric or symbolic variables. The number of output variables or symbolic arrays must equal the number of variables to be solved for. If you explicitly specify independent variables vars, then the solver uses the same order to return the solutions. If you do not specify vars, the toolbox sorts independent variables alphabetically, and then assigns the solutions for these variables to the output variables or symbolic arrays.

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### Tips

• vpasolve returns all solutions only for polynomial equations. For nonpolynomial equations, there is no general method of finding all solutions. When you look for numerical solutions of a nonpolynomial equation or system that has several solutions, then, by default, vpasolve returns only one solution, if any. To find more than just one solution, set random to true. Now, calling vpasolve repeatedly might return several different solutions.

• When you solve a system where there are not enough equations to determine all variables uniquely, the behavior of vpasolve behavior depends on whether the system is polynomial or nonpolynomial. If polynomial, vpasolve returns all solutions by introducing an arbitrary parameter. If nonpolynomial, a single numerical solution is returned, if it exists.

• When you solve a system of rational equations, the toolbox transforms it to a polynomial system by multiplying out the denominators. vpasolve returns all solutions of the resulting polynomial system, including those that are also roots of these denominators.

• vpasolve ignores assumptions set on variables. You can restrict the returned results to particular ranges by specifying appropriate search ranges using the argument init_guess.

• If init_guess specifies 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.

• The output variables y1,...,yN do not specify the variables for which vpasolve solves equations or systems. If y1,...,yN are the variables that appear in eqns, that does not guarantee that vpasolve(eqns) will assign the solutions to y1,...,yN using the correct order. Thus, for the call [a,b] = vpasolve(eqns), you might get the solutions for a assigned to b and vice versa.

To ensure the order of the returned solutions, specify the variables vars. For example, the call [b,a] = vpasolve(eqns,b,a) assigns the solutions for a assigned to a and the solutions for b assigned to b.

• Place equations and expressions to the left of the argument list, and the variables to the right. vpasolve checks for variables starting on the right, and on reaching the first equation or expression, assumes everything to the left is an equation or expression.

• If possible, solve equations symbolically using solve, and then approximate the obtained symbolic results numerically using vpa. Using this approach, you get numeric approximations of all solutions found by the symbolic solver. Using the symbolic solver and postprocessing its results requires more time than using the numeric methods directly. This can significantly decrease performance.

### Algorithms

• When you set random to true and specify a search range for a variable, random starting points within the search range are chosen using the internal random number generator. The distribution of starting points within finite search ranges is uniform.

• When you set random to true and do not specify a search range for a variable, random starting points are generated using a Cauchy distribution with a half-width of 100. This means the starting points are real valued and have a large spread of values on repeated calls.