“simultaneous equations”的概念、定义、翻译、参考文献-科学参考

simultaneous equations

Computer

A set of equations that together define an unknown set of values or functions. The term is normally applied to linear algebraic equations.

Electronics and Electrical Engineering

A set of equations used in the solution of n linear algebraic equations in n unknowns, having the general form:
$\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + a_{13} x_{3} + \dots + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + a_{23} x_{3} + \dots + a_{2 n} x_{n} = b_{2} \\ a_{31} x_{1} + a_{32} x_{2} + a_{33} x_{3} + \dots + a_{3 n} x_{n} = b_{3} \\ \dots \dots \\ a_{n 1} x_{1} + a_{n 2} x_{2} + a_{n 3} x_{3} + \dots + a_{n n} x_{n} = b \end{array}$
Cramer’s rule states that the jth unknown is given directly by
$x_{j} = \frac{Δ_{j}}{Δ}$
where Δ is the determinant of the matrix of the coefficients a_ij (i,j = 1, 2, 3,…,n) and Δ_j is the determinant of the matrix obtained from the matrix of the coefficients a_ij by replacing the coefficients of the jth column with the coefficients b_j (j = 1, 2, 3,…,n). This assumes that the equations are linearly independent, i.e. that Δ ≠ 0.
The Gaussian elimination technique repeatedly combines different rows such that the original system of equations is transformed into a system of the type shown below. This is achieved by taking the first equation of the original system and multiplying it by a factor such that when added to the second equation the coefficient of x₁ is zero. This is repeated but with the third equation and so on until all the coefficients of x₁ become zero. With this new set of equations the technique is repeated so that the coefficients of x₂ become zero. This is repeated until x_n can be solved directly, i.e. α_nnx_n = β_n. Then by a process of back-substitution all the other unknowns can be determined.
$\begin{matrix} α_{11} x_{1} + α_{12} x_{2} + α_{13} x_{3} + \dots + α_{1 n} x_{n} = β_{1} \\ α_{22} x_{2} + α_{23} x_{3} + \dots + α_{2 n} x_{n} = β_{2} \\ α_{33} x_{3} + \dots + α_{3 n} x_{n} = β_{3} \\ \dots \dots \\ α_{n n} x_{n} = β_{n} \end{matrix}$
Gaussian elimination and Cramer’s rule are used extensively in circuit analysis, where the simultaneous equations are formulated using either node or loop analysis: x₁ through x_n are the unknown node voltages (or mesh currents), a_ij (i,j = 1, 2, 3…,n) are known admittances (or resistances), and b₁ through b_n are known source voltages (or currents).

单词	simultaneous equations
释义	simultaneous equations Computer A set of equations that together define an unknown set of values or functions. The term is normally applied to linear algebraic equations. Electronics and Electrical Engineering A set of equations used in the solution of n linear algebraic equations in n unknowns, having the general form: $\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + a_{13} x_{3} + \dots + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + a_{23} x_{3} + \dots + a_{2 n} x_{n} = b_{2} \\ a_{31} x_{1} + a_{32} x_{2} + a_{33} x_{3} + \dots + a_{3 n} x_{n} = b_{3} \\ \dots \dots \\ a_{n 1} x_{1} + a_{n 2} x_{2} + a_{n 3} x_{3} + \dots + a_{n n} x_{n} = b \end{array}$ Cramer’s rule states that the jth unknown is given directly by $x_{j} = \frac{Δ_{j}}{Δ}$ where Δ is the determinant of the matrix of the coefficients a_ij (i,j = 1, 2, 3,…,n) and Δ_j is the determinant of the matrix obtained from the matrix of the coefficients a_ij by replacing the coefficients of the jth column with the coefficients b_j (j = 1, 2, 3,…,n). This assumes that the equations are linearly independent, i.e. that Δ ≠ 0. The Gaussian elimination technique repeatedly combines different rows such that the original system of equations is transformed into a system of the type shown below. This is achieved by taking the first equation of the original system and multiplying it by a factor such that when added to the second equation the coefficient of x₁ is zero. This is repeated but with the third equation and so on until all the coefficients of x₁ become zero. With this new set of equations the technique is repeated so that the coefficients of x₂ become zero. This is repeated until x_n can be solved directly, i.e. α_nnx_n = β_n. Then by a process of back-substitution all the other unknowns can be determined. $\begin{matrix} α_{11} x_{1} + α_{12} x_{2} + α_{13} x_{3} + \dots + α_{1 n} x_{n} = β_{1} \\ α_{22} x_{2} + α_{23} x_{3} + \dots + α_{2 n} x_{n} = β_{2} \\ α_{33} x_{3} + \dots + α_{3 n} x_{n} = β_{3} \\ \dots \dots \\ α_{n n} x_{n} = β_{n} \end{matrix}$ Gaussian elimination and Cramer’s rule are used extensively in circuit analysis, where the simultaneous equations are formulated using either node or loop analysis: x₁ through x_n are the unknown node voltages (or mesh currents), a_ij (i,j = 1, 2, 3…,n) are known admittances (or resistances), and b₁ through b_n are known source voltages (or currents).
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