方程式stomatrix

Convert linear equations to matrix form

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Syntax

[A,b] = equationsToMatrix(eqns)

[A,b] = equationsToMatrix(eqns,vars)

a = equationStomatrix（___)

Description

example

[A,b] = equationStomatrix（eqns)converts equationseqns到矩阵形式。eqns必须是所有变量中方程式的线性系统symvarfinds ineqns。

example

[A,b] = equationStomatrix（eqns,vars)convertseqnsto matrix form, whereeqns必须是线性的vars。

example

A= equationsToMatrix(___)仅返回方程系统的系数矩阵。

Examples

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Convert Linear Equations to Matrix Form

打开实时脚本

Convert a system of linear equations to matrix form.方程式stomatrixautomatically detects the variables in the equations by usingsymvar。The returned coefficient matrix follows the variable order determined bysymvar。

symsxyzeqns = [x+y-2*z == 0, x+y+z == 1, 2*y-z == -5]; [A,b] = equationsToMatrix(eqns)

A =
                      
                        $(\begin{array}{c} 1 & 1 & - 2 \\ 1 & 1 & 1 \\ 0 & 2 & - 1 \end{array})$

b =
                      
                        $(\begin{array}{c} 0 \\ 1 \\ - 5 \end{array})$

vars =symvar(eqns)

vars =
                       $(\begin{array}{c} x & y & z \end{array})$

You can change the arrangement of the coefficient matrix by specifying other variable order.

vars =[x,z,y]; [A,b] = equationsToMatrix(eqns,vars)

A =
                      
                        $(\begin{array}{c} 1 & - 2 & 1 \\ 1 & 1 & 1 \\ 0 & - 1 & 2 \end{array})$

b =
                      
                        $(\begin{array}{c} 0 \\ 1 \\ - 5 \end{array})$

Specify Variables in Equations

打开实时脚本

Convert a linear system of equations to the matrix form by specifying independent variables. This is useful when the equation are only linear in some variables.

For this system, specify the variables as[s t]因为系统不线性r。

symsrsteqns = [s-2*t+r^2 == -1 3*s-t == 10]; vars = [s t]; [A,b] = equationsToMatrix(eqns,vars)

A =
                      
                        $(\begin{array}{c} 1 & - 2 \\ 3 & - 1 \end{array})$

b =
                      
                        $(\begin{array}{c} - r^{2} - 1 \\ 10 \end{array})$

只返回系数后来x of Equations

打开实时脚本

Return only the coefficient matrix of the equations by specifying a single output argument.

symsxyzeqns = [x+y-2*z == 0, x+y+z == 1, 2*y-z == -5]; vars = [x y z]; A = equationsToMatrix(eqns,vars)

A =
                      
                        $(\begin{array}{c} 1 & 1 & - 2 \\ 1 & 1 & 1 \\ 0 & 2 & - 1 \end{array})$

Solve System of Equations That Are Functions of Time

打开实时脚本

考虑以下时间函数的线性方程系统：

$\begin{array}{l} 2 x (t) + y (t) + z (t) = 2 u (t) \\ - x (t) + y (t) - z (t) = v (t) \\ x (t) + 2 y (t) + 3 z (t) = - 10 \end{array}$

Declare the system of equations.

symsx(t)y(t)z（t）u（t）v(t)eqn1 = 2*x + y + z == 2*u; eqn2 = -x + y - z == v; eqn3 = x + 2*y + 3*z == -10; eqn = [eqn1; eqn2; eqn3]

eqn(t) =
                      
                        $(\begin{array}{c} 2 x (t) + y (t) + z (t) = 2 u (t) \\ y (t) - x (t) - z (t) = v (t) \\ x (t) + 2 y (t) + 3 z (t) = - 10 \end{array})$

Specify the independent variables $x (t)$ , $y (t)$ ，和 $z (t)$ 在方程式中作为符号向量vars。Use the方程式stomatrixfunction to convert the system of equations into the matrix form.

vars =[x(t); y(t); z(t)]; [A,b] = equationsToMatrix(eqn,vars)

A =
                      
                        $(\begin{array}{c} 2 & 1 & 1 \\ - 1 & 1 & - 1 \\ 1 & 2 & 3 \end{array})$

b =
                      
                        $(\begin{array}{c} 2 u (t) \\ v (t) \\ - 10 \end{array})$

Solve the matrix form of the equations using thelinsolvefunction.

X = linsolve(A,b)

X =
                      
                        $(\begin{array}{c} \frac{10 u (t)}{9} - \frac{v (t)}{9} + \frac{20}{9} \\ \frac{4 u (t)}{9} + \frac{5 v (t)}{9} - \frac{10}{9} \\ - \frac{2 u (t)}{3} - \frac{v (t)}{3} - \frac{10}{3} \end{array})$

Evaluate the $z (t)$ solution for the functions $u (t) = \cos (t)$ 和 $v (t) = \sin (2 t)$ 。绘制 $z (t)$ solution.

ZSOL =subs(X(3),[u(t) v(t)],[cos(t) sin(2*t)])

ZSOL =
                      
                        $- \frac{\sin (2 t)}{3} - \frac{2 \cos (t)}{3} - \frac{10}{3}$

fplot(zSol)

图包含一个轴对象。轴对象包含一个类型函数线的对象。

Input Arguments

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`eqns`—Linear equations
符号方程或表达式的向量

Linear equations, specified as a vector of symbolic equations or expressions. Symbolic equations are defined by using the==operator, such asx + y == 1。For symbolic expressions,方程式stomatrixassumes that the right side is 0.

Equations must be linear in terms ofvars。

`vars`—Independent variables
vector of symbolic variables(default) |vector of symbolic functions

Independent variables ineqns, specified as a vector of symbolic variables or symbolic functions.

Output Arguments

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`A`— Coefficient matrix
symbolic matrix

Coefficient matrix of the system of linear equations, specified as a symbolic matrix.

`b`— Right sides of equations
symbolic matrix

Vector containing the right sides of equations, specified as a symbolic matrix.

More About

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线性方程系统的矩阵表示

线性方程系统

$\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} \\ \dots \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} = b_{m} \end{array}$

can be represented as the matrix equation $A \cdot \vec{x} = \vec{b}$ 。这里，Ais the coefficient matrix.

$A = (\begin{matrix} a_{11} & \dots & a_{1 n} \\ ⋮ & ⋱ & ⋮ \\ a_{m 1} & \dots & a_{m n} \end{matrix})$

$\vec{b}$ 是包含方程式右侧的向量。

$\vec{b} = (\begin{matrix} b_{1} \\ ⋮ \\ b_{m} \end{matrix})$

方程式stomatrix

Syntax

Description

Examples

Convert Linear Equations to Matrix Form

Specify Variables in Equations

只返回系数后来x of Equations

Solve System of Equations That Are Functions of Time

Input Arguments

`eqns`—Linear equations
符号方程或表达式的向量

`vars`—Independent variables
vector of symbolic variables(default) |vector of symbolic functions

Output Arguments

`A`— Coefficient matrix
symbolic matrix

`b`— Right sides of equations
symbolic matrix

More About

线性方程系统的矩阵表示

See Also

Topics

符号数学工具箱Documentation

万博1manbetx

Mathematical Modeling with Symbolic Math Toolbox

方程式stomatrix

Syntax

Description

Examples

Convert Linear Equations to Matrix Form

Specify Variables in Equations

只返回系数后来x of Equations

Solve System of Equations That Are Functions of Time

Input Arguments

eqns—Linear equations符号方程或表达式的向量

vars—Independent variablesvector of symbolic variables(default) |vector of symbolic functions

Output Arguments

A— Coefficient matrixsymbolic matrix

b— Right sides of equationssymbolic matrix

More About

线性方程系统的矩阵表示

See Also

Topics

符号数学工具箱Documentation

万博1manbetx

Mathematical Modeling with Symbolic Math Toolbox

`eqns`—Linear equations
符号方程或表达式的向量

`vars`—Independent variables
vector of symbolic variables(default) |vector of symbolic functions

`A`— Coefficient matrix
symbolic matrix

`b`— Right sides of equations
symbolic matrix