 |
BUILT-IN MATHEMATICA SYMBOL
Solve
Solve[expr, vars]
attempts to solve the system expr of equations or inequalities for the variables vars.
Details and Options
- The system expr can be any logical combination of:
-
lhs==rhs equations lhs!=rhs inequations 
or 
inequalities expr 
domdomain specifications ForAll[x,cond,expr] universal quantifiers Exists[x,cond,expr] existential quantifiers - Solve[{expr1, expr2, ...}, vars] is equivalent to Solve[expr1&&expr2&&..., vars].
- A single variable or a list of variables can be specified.
- Solve gives solutions in terms of rules of the form
. - When there are several variables, the solution is given in terms of lists of rules:
. - When there are several solutions, Solve gives a list of them.
- When a single variable is specified and a particular root of an equation has multiplicity greater than one, Solve gives several copies of the corresponding solution.
- Solve[expr, vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
- Solve[expr, vars, dom] restricts all variables and parameters to belong to the domain dom.
- If dom is Reals, or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.
- Solve[expr&&varsReals, vars, Complexes] solves for real values of variables, but function values are allowed to be complex.
- Solve[expr, vars, Integers] solves Diophantine equations over the integers.
- Algebraic variables in expr free of the
and of each other are treated as independent parameters. - Solve deals primarily with linear and polynomial equations.
- When expr involves only polynomial equations and inequalities over real or complex domains, then Solve can always in principle solve directly for all the
. - When expr involves transcendental conditions or integer domains, Solve will often introduce additional parameters in its results.
- Solve can give explicit representations for solutions to all linear equations and inequalities over the integers, and can solve a large fraction of Diophantine equations described in the literature.
- When expr involves only polynomial conditions over real or complex domains, Solve[expr, vars] will always be able to eliminate quantifiers.
- The following options can be given:
-
Cubics False whether to use explicit radicals to solve all cubics GeneratedParameters C how to name parameters that are generated InverseFunctions Automatic whether to use symbolic inverse functions MaxExtraConditions 0 how many extra equational conditions on continuous parameters to allow Method Automatic what method should be used Modulus 0 modulus to assume for integers Quartics False whether to use explicit radicals to solve all quartics VerifySolutions Automatic whether to verify solutions obtained using non-equivalent transformations WorkingPrecision Infinity precision to be used in computations - Solve gives generic solutions only. Solutions that are valid only when continuous parameters satisfy equations are removed. Other solutions that are only conditionally valid are expressed as ConditionalExpression objects.
- Conditions included in ConditionalExpression solutions may involve inequalities, Element statements, equations and inequations on non-continuous parameters, and equations with full-dimensional solutions. Inequations and NotElement conditions on continuous parameters and variables are dropped.
- With MaxExtraConditions->Automatic, only solutions that require the minimal number of equational conditions on continuous parameters are included.
- With MaxExtraConditions->All, solutions that require arbitrary conditions on parameters are given and all conditions are included.
- With MaxExtraConditions->k, only solutions that require at most k equational conditions on continuous parameters are included.
- Solve uses non-equivalent transformations to find solutions of transcendental equations and hence it may not find some solutions and may not establish exact conditions on the validity of the solutions found.
- With Method->Reduce, Solve uses only equivalent transformations and finds all solutions.
- Solve gives
if there are no possible solutions to the equations. - Solve gives
if the set of solutions is full-dimensional. - Solve[eqns, ..., Modulus->m] solves equations over the integers modulo m. With Modulus->Automatic, Solve will attempt to find the largest modulus for which the equations have solutions.
- Solve uses special efficient techniques for handling sparse systems of linear equations with approximate numerical coefficients.
Examplesopen all
Basic Examples (6)
| In[1]:= |
| Out[1]= |  |
Solve simultaneous equations in 
and 
:
| In[1]:= |
| Out[1]= |  |
Solutions are given as lists of replacements:
| In[1]:= |
| Out[1]= |  |
| In[2]:= |
| Out[2]= |  |
Replace combinations of 
and 
by solutions, and simplify the results:
| In[3]:= |
| Out[3]= |  |
Plot the real parts of the solutions for 
as a function of the parameter 
:
| In[4]:= |
| Out[4]= |  |
| In[1]:= |
| Out[1]= |  |
| In[2]:= |
| Out[2]= |  |
Solve an equation over the reals:
| In[1]:= |
| Out[1]= |  |
Replace 
by solutions and simplify the results:
| In[2]:= |
| Out[2]= |  |
Solve an equation over the positive integers:
| In[1]:= |
| Out[1]= |  |
| In[2]:= |
| Out[2]= |  |
New in 1 | Last modified in 8
