Documentation

linsolve

Solve a system of linear equations

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Syntax

linsolve(eqs, options)
linsolve(eqs, vars, options)

Description

linsolve(eqs, vars) solves a system of linear equations with respect to the unknowns vars.

linsolve(eqs, < vars , < ShowAssumptions >>) solves the linear system eqs with respect to the unknowns vars. If no unknowns are specified, then linsolve solves for all indeterminates in eqs; the unknowns are determined internally by indets(eqs,PolyExpr).

linsolve(eqs, vars, Domain = R) solves the system over the domain R, which must be a field, i.e., a domain of category Cat::Field.

Note

Note that the return format does not allow to return kernel elements if elements of the domain R cannot be multiplied with the symbolic unknowns that span the kernel. In such a case, linsolve issues a warning and returns only a special solution. The kernel can be computed via linalg::matlinsolve for any field R.

Each element of eqs must be either an equation or an arithmetical expression f, which is considered to be equivalent to the equation f = 0.

The unknowns in vars need not be identifiers or indexed identifiers; expressions such as sin(x), f(x), or y^(1/3) are allowed as well. More generally, any expression accepted as indeterminate by poly is a valid unknown.

If the option ShowAssumptions is not given and the system is solvable, then the return value is a list of equations of the form var = value, where var is one of the unknowns in vars and value is an arithmetical expression that does not involve any of the unknowns on the left side of a returned equation. Note that if the solution manifold has dimension greater than zero, then some of the unknowns in vars will occur on the right side of some returned equations, representing the degrees of freedom. See Example 2.

If vars is a list, then the solved equations are returned in the the same order as the unknowns in vars.

The function linsolve can only solve systems of linear equations. Use solve for nonlinear equations.

linsolve is an interface function to the procedures numeric::linsolve and linalg::matlinsolve. For more details see the numeric::linsolve, linalg::matlinsolve help pages and the background section of this help page.

The system eqs is checked for linearity. Since such a test can be expensive, it is recommended to use numeric::linsolve or linalg::matlinsolve directly when you know that the system is linear.

Note

linsolve does not react to properties of identifiers set by assume.

Examples

Example 1

Equations and variables may be entered as sets or lists:

linsolve({x + y = 1, 2*x + y = 3}, {x, y}),
linsolve({x + y = 1, 2*x + y = 3}, [x, y]),
linsolve([x + y = 1, 2*x + y = 3], {x, y}),
linsolve([x + y = 1, 2*x + y = 3], [x, y]) Also expressions may be used as variables:

linsolve({cos(x) + sin(x) = 1, cos(x) - sin(x) = 0},
{cos(x), sin(x)}) Furthermore, indexed identifiers are valid, too:

S := linsolve({2*a + 3*a = 5, 7*a + 11*a = 13,
17*a + 19*a = 23}, {a, a, a}) Assign individual solutions to variables using assign. Alternatively, access the solution by indexing into S:

assign(S):
a;
a2_val := S;  Delete a for use in further computations.

delete a;

Next, we demonstrate the use of option Domain and solve a system over the field 23 with it:

linsolve([2*x + y = 1, -x - y = 0],
Domain = Dom::IntegerMod(23)) The following system does not have a solution:

linsolve({x + y = 1, 2*x + 2*y = 3}, {x, y}) Example 2

If the solution of the linear system is not unique, then some of the unknowns are used as “free parameters” spanning the solution space. In the following example the unknown z is such a parameter. It does not appear on the left side of the solved equations:

eqs := [x + y = z, x + 2*y = 0, 2*x - z = -3*y, y + z = 0]:
vars := [w, x, y, z]:
linsolve(eqs, vars) Example 3

If you use the Normal option, linsolve calls the normal function for final results. This call ensures that linsolve returns results in normalized form:

linsolve([x + a*y = a + 1, b*x - y = b - 1], {x, y}) If you specify Normal = FALSE, linsolve does not call normal for the final result:

linsolve([x + a*y = a + 1, b*x - y = b - 1], {x, y}, Normal = FALSE) Example 4

Solve this system:

eqs := [x + a*y = b, x + A*y = b]:
linsolve(eqs, [x, y]) Note that more solutions exist for a = A. linsolve omits these solutions because it makes some additional assumptions on symbolic parameters of this system. To see the assumptions that linsolve made while solving this system, use the ShowAssumptions option:

linsolve(eqs, [x, y], ShowAssumptions) delete eqs:

Parameters

 eqs A list or a set of linear equations or arithmetical expressions vars A list or a set of unknowns to solve for: typically identifiers or indexed identifiers

Options

 Domain Option, specified as Domain = R Solve the system over the field R, which must be a domain of category Cat::Field. Normal Option, specified as Normal = b Return normalized results. The value b must be TRUE or FALSE. By default, Normal = TRUE, meaning that linsolve guarantees normalization of the returned results. Normalizing results can be computationally expensive. By default, linsolve calls normal before returning results. This option affects the output only if the solution contains variables or exact expressions, such as sqrt(5) or sin(PI/7). To avoid this additional call, specify Normal = FALSE. In this case, linsolve also can return normalized results, but does not guarantee such normalization. See Example 3. ShowAssumptions Return information about internal assumptions that linsolve made on symbolic parameters in eqs. With this option, linsolve returns a list [Solution, Constraints, Pivots]. Solution is a list of solved equations representing the complete solution manifold of eqs, as described above. The lists Constraints and Pivots contain equations and inequalities involving symbolic parameters in eqs. Internally, these were assumed to hold true when solving the system. See Example 4. When Gaussian elimination produces an equation 0 = c with nonzero c, linsolve without ShowAssumptions returns FAIL. If c involves symbolic parameters, try using linsolve with ShowAssumptions to solve such systems. If the system is solvable, you will get the solution. In this case, an equation 0 = c is returned in the Constraints list. If the system is not solvable, linsolve with ShowAssumptions returns [FAIL, [], []].

Return Values

Without the ShowAssumptions option, a list of simplified equations is returned. It represents the general solution of the system eqs. FAIL is returned if the system is not solvable.

With ShowAssumptions, a list [Solution, Constraints, Pivots] is returned. Solution is a list of simplified equations representing the general solution of eqs. The lists Constraints and Pivots contain equations and inequalities involving symbolic parameters in eqs. Internally, these were assumed to hold true when solving the system.

Algorithms

If the option Domain is not present, the system is solved by calling numeric::linsolve with the option Symbolic.

If the option Domain = R is given and R is either Dom::ExpressionField() or Dom::Float, then numeric::linsolve is used to compute the solution of the system. This function uses a sparse representation of the equations.

Otherwise, eqs is first converted into a matrix and then solved by linalg::matlinsolve. A possibly sparse structure of the input system is not taken into account.