Casoratian
Casoratian
Casoratian[{y1,y2,…},n]
给出关于 n 的序列 y1, y2, … 的 Casoratian 行列式.
Casoratian[eqn,y,n]
给出关于 y[n+m] 的线性微分方程 eqn 的解基的 Casoratian 行列式.
Casoratian[eqns,{y1,y2,…},n]
给出线性微分方程组 eqns 的 Casoratian 行列式.
更多信息和选项
- Casoratian 行列式定义为：Det[Table[DiscreteShift[yi,{n,j}],{i,m},{j,0,m-1}]].
- 如果 y1,y2,… 线性相关，则 Casoratian 在各处消失.
范例
打开所有单元 关闭所有单元基本范例 (4)
Casoratian[{2 ^ n, 3 ^ n}, n]Casoratian[{2 ^ n, n 2 ^ n}, n]Casoratian[{2 ^ n, 2 ^ (n + 3)}, n]Casoratian[{n ^ 2, 3n ^ 2 + 5n + 1, n ^ 2 + 5n + 1}, n]
Casoratian[{n ^ 2, n ^ 2 + a}, n]Casoratian[y[n + 2] == y[n + 1] + y[n], y, n]RSolve[y[n + 2] == y[n + 1] + y[n], y, n]Casoratian[{Fibonacci[n], LucasL[n]}, n]范围 (9)
序列 (5)
Casoratian[{n, n ^ 2}, n]Casoratian[{n, n ^ 2, n ^ 3}, n]Casoratian[{(1/n), (1/n + 1)}, n]Casoratian[{(1/n), (1/n + 1), (1/n(n + 1))}, n]Casoratian[{3 ^ n, 3 ^ (n + 4)}, n]Casoratian[{3 ^ n, n 3 ^ n}, n]Casoratian[{Fibonacci[n], LucasL[n]}, n]Casoratian[{Sin[n 2Pi / 3], Cos[n 2Pi / n]}, n]Casoratian[{n!, (n + 1)!}, n]DiscretePlot[Log[%], {n, 10}]微分方程 (4)
Casoratian[y[n + 3] - 5y[n + 2] + 11y[n] == 0, y, n]RSolve[y[n + 3] - 5y[n + 2] + 11y[n] == 0, y, n]Casoratian[Coefficient[y[n] /. %[[1]], Table[C[i], {i, 3}]], n]cas = Casoratian[y[n + 2] - (n + 5)y[n] == 0, y, n]RSolve[y[n + 2] - (n + 5)y[n] == 0, y, n]Casoratian[Coefficient[y[n] /. %[[1]], {C[1], C[2] }], n]Table[% / cas, {n, 1, 7}]//UnionCasoratian[y[n + 2] - n!y[n] == 0, y, n]Casoratian[y[n + 100] - y[n + 99] + Binomial[2n, n]y[n] == 0, y, n]Casoratian[y[n + 2] - (Mod[n, 3] + 1)y[n] == 0, y, n]Casoratian[y[n + 2] - Sin[n 2Pi / 5 + Pi / 3]y[n] == 0, y, n]Casoratian[y[n + 2] - n a ^ n Sin[n 2Pi / 5 + Pi / 3]y[n] == 0, y, n]应用 (2)
VariationOfParameters[homogeneouseqn_, forcingterm_, y_, n_] :=
Block[{sol, r, y1, y2, c, u1, u2},
sol = RSolve[homogeneouseqn, y, n];
y1[r_] := (y[r] /. sol[[1]] /. {C[1] -> 1, C[2] -> 0});
y2[r_] := (y[r] /. sol[[1]] /. {C[1] -> 0, C[2] -> 1});
c[r_] := Casoratian[{y1[r], y2[r]}, r];
u1 = -Sum[y2[n + 1] forcingterm / c[n + 1], {n, 0, n - 1}];
u2 = Sum[y1[n + 1] forcingterm / c[n + 1], {n, 0, n - 1}];
Simplify[y1[n] u1 + y2[n] u2, Element[n, Integers]]
]VariationOfParameters[y[n + 2] - 4y[n] == 0, n ^ 2, y, n]RSolve[y[n + 3] + y[n] == 0, y, n]Casoratian[Coefficient[y[n] /. %[[1]], Table[C[i], {i, 3}]], n]属性和关系 (6)
Casoratian 等价于一个行列式：
Casoratian[{f[n], g[n]}, n]Det[Table[DiscreteShift[{f[n], g[n]}, {n, m}], {m, 0, 1}]]Casoratian 检测线性相关：
Casoratian[{f[n], c f[n]}, n]Casoratian[{f[n] + g[n], a f[n] + b g[n]}, n]Wronskian 执行连续参数函数的线性相关：
Wronskian[{Sin[x], Exp[x]}, x]Wronskian[{Sin[x], Sin[2x]}, x]Plot[{Sin[x], Sin[2x]}, {x, 0, 8Pi}]{Sin[x], Sin[2x]} /. x -> n Pi / 2Casoratian[%, n]DiscretePlot[{Sin[x], Sin[2x]}, {x, 0, 8Pi, Pi / 2}]用 Orthogonalize 产生线性无关序列的集合：
basis = Orthogonalize[{1, n, n ^ 2}, Sum[#1 #2 (1 / 2) ^ n, {n, 0, Infinity}]&]coef = Sum[(n + 1)(n + 2)basis (1 / 2) ^ n, {n, 0, ∞}]{coef.basis, (n + 1)(n + 2)}//ExpandOrthogonalize[{1, n, n ^ 2, (n + 1)(n + 2)}, Sum[#1 #2 (1 / 2) ^ n, {n, 0, Infinity}]&]用 Reduce 将多项式和有理序列用对方表示：
Reduce[ForAll[n, a + b n + c n ^ 2 == (n + 1)(n + 2)], {a, b, c}]Reduce[ForAll[n, n ≠ -1 && n ≠ 0, (a/n) + (b/n + 1) == (1/n(n + 1))], {a, b}]
Wolfram Research (2008)，Casoratian，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Casoratian.html.
文本
Wolfram Research (2008)，Casoratian，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Casoratian.html.
CMS
Wolfram 语言. 2008. "Casoratian." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/Casoratian.html.
APA
Wolfram 语言. (2008). Casoratian. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/Casoratian.html 年