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帕德近似

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近似值透過將函式展開為兩個冪級數的比率，並確定分子和分母係數而匯出。當函式包含極點時，帕德近似通常優於泰勒級數，因為有理函式的使用允許它們得到很好的表示。

帕德近似 R_(L/0) 對應於麥克勞林級數。當它存在時，任何冪級數的 R_(L/M)=[L/M] 帕德近似是

A(x)=sum_(j=0)^inftya_jx^j
(1)

唯一的。如果 A(x) 是一個超越函式，則這些項由關於 x_0 的泰勒級數給出

a_n=1/(n!)A^((n))(x_0).
(2)

係數透過設定以下公式找到

A(x)-(P_L(x))/(Q_M(x))=0
(3)

並等同係數。Q_M(x) 可以乘以任意常數，這將重新調整其他係數，因此可以應用額外的約束。傳統的歸一化是

Q_M(0)=1.
(4)

展開 (3) 得到

P_L(x)=p_0+p_1x+...+p_Lx^L
(5)
Q_M(x)=1+q_1x+...+q_Mx^M.
(6)

這些給出了方程組

a_0=p_0
(7)
a_1+a_0q_1=p_1
(8)
a_2+a_1q_1+a_0q_2=p_2
(9)
|
(10)
a_L+a_(L-1)q_1+...+a_0q_L=p_L
(11)
a_(L+1)+a_Lq_1+...+a_(L-M+1)q_M=0
(12)
|
(13)
a_(L+M)+a_(L+M-1)q_1+...+a_Lq_M=0,
(14)

其中 a_n=0 對於 n<0 且 q_j=0 對於 j>M。直接求解這些得到

[L/M]=(|a_(L-m+1) a_(L-m+2) ... a_(L+1); | | ... |; a_L a_(L+1) ... a_(L+M); sum_(j=M)^(L)a_(j-M)x^j sum_(j=M-1)^(L)a_(j-M+1)x^j ... sum_(j=0)^(L)a_jx^j|)/(|a_(L-M+1) a_(L-M+2) ... a_(L+1); | | ... |; a_L a_(L+1) ... a_(L+M); x^M x^(M-1) ... 1|),
(15)

其中如果下索引超過上索引，則和替換為零。另種形式為

[L/M]=sum_(j=0)^(L-M)a_jx^j+x^(L-M+1)w_(L/M)^(T)W_(L/M)^(-1)w_(L/M) =sum_(j=0)^(L+n)a_jx^j+x^(L+n+1)w_((L+M)/M)^TW_(L/M)^(-1)w_((L+n)/M)
(16)

對於

W_(L/M)=[a_(L-M+1)-xa_(L-M+2) ... a_L-xa_(L+1); | ... |; a_L-xa_(L+1) ... a_(L+M-1)-xa_(L+M)]
(17)
w_(L/M)=[a_(L-M+1); a_(L-M+2); |; a_L],
(18)

和 0<=n<=M。

例如，e^x 的前幾個帕德近似是

exp_(0/0)(x)=1
(19)
exp_(0/1)(x)=1/(1-x)
(20)
exp_(0/2)(x)=2/(2-2x+x^2)
(21)
exp_(0/3)(x)=6/(6-6x+3x^2-x^3)
(22)
exp_(1/0)(x)=1+x
(23)
exp_(1/1)(x)=(2+x)/(2-x)
(24)
exp_(1/2)(x)=(6+2x)/(6-4x+x^2)
(25)
exp_(1/3)(x)=(24+6x)/(24-18x+6x^2-x^3)
(26)
exp_(2/0)(x)=(2+2x+x^2)/2
(27)
exp_(2/1)(x)=(6+4x+x^2)/(6-2x)
(28)
exp_(2/2)(x)=(12+6x+x^2)/(12-6x+x^2)
(29)
exp_(2/3)(x)=(60+24x+3x^2)/(60-36x+9x^2-x^3)
(30)
exp_(3/0)(x)=(6+6x+3x^2+x^3)/6
(31)
exp_(3/1)(x)=(24+18x+6x^2+x^3)/(24-6x)
(32)
exp_(3/2)(x)=(60+36x+9x^2+x^3)/(60-24x+3x^2)
(33)
exp_(3/3)(x)=(120+60x+12x^2+x^3)/(120-60x+12x^2-x^3).
(34)

兩項恆等式包括

(P_(L+1)(x))/(Q_(M+1)(x))-(P_L^'(x))/(Q_M^'(x))=(C_((L+1)/(M+1))^2x^(L+M+1))/(Q_(M+1)(x)Q_M^'(x))
(35)
(P_(L+1)(x))/(Q_M(x))-(P_L^'(x))/(Q_M^'(x))=(C_((L+1)/M)C_((L+1)/(M+1))x^(L+M+1))/(Q_M(x)Q_M^'(x))
(36)
(P_L(x))/(Q_(M+1)(x))-(P_L^'(x))/(Q_M^'(x))=(C_(L/(M+1))C_((L+1)/(M+1))x^(L+M+1))/(Q_M(x)Q_M^'(x))
(37)
(P_L(x))/(Q_(M+1)(x))-(P_(L+1)^'(x))/(Q_M^')=(C_((L+1)/(M+1))^2x^(L+M+2))/(Q_(M+1)Q_M^')
(38)
(P_(L+1))/(Q_M(x))-(P_(L-1)^'(x))/(Q_M^'(x))=(C_(L/(M+1))C_((L+1)/M)x^(L+M)+C_(L/M)C_((L+1)/(M+1))x^(L+M+1))/(Q_M(x)Q_M^'(x))
(39)
(P_L(x))/(Q_(M+1)(x))-(P_L^'(x))/(Q_(M-1)^'(x))=(C_(L/(M+1))C_((L+1)/M)x^(L+M)-C_(L/M)C_((L+1)/(M+1))x^(L+M+1))/(Q_(M+1)(x)Q_(M-1)^'(x)),
(40)

其中 C 是 C-行列式。可以使用弗羅貝尼烏斯三角形恆等式推匯出三項恆等式 (Baker 1975, p. 32)。

五項恆等式是

S_((L+1)/M)S_((L-1)/M)-S_(L/(M+1))S_(L/(M-1))=S_(L/M)^2.
(41)

交比恆等式包括

((R_(L/M)-R_(L/(M+1)))(R_((L+1)/M)-R_((L+1)/(M+1))))/((R_(L/M)-R_((L+1)/M))(R_(L/(M+1))-R_((L+1)/(M+1))))=(C_(L/(M+1))C_((L+2)/(M+1)))/(C_((L+1)/M)C_((L+1)/(M+2)))
(42)
((R_(L/M)-R_((L+1)/(M+1)))(R_((L+1)/M)-R_(L/(M+1))))/((R_(L/M)-R_(L/(M+1)))(R_((L+1)/M)-R_((L+1)/(M+1))))=(C_((L+1)/(M+1))^2x)/(C_(L/(M+1))C_((L+2)/(M+1)))
(43)
((R_(L/M)-R_((L+1)/(M+1)))(R_((L+1)/M)-R_(L/(M+1))))/((R_(L/M)-R_((L+1)/M))(R_(L/(M+1))-R_((L+1)/(M+1))))=(C_((L+1)/(M+1))^2x)/(C_((L+1)/M)C_((L+1)/(M+2)))
(44)
((R_(L/M)-R_((L+1)/(M-1)))(R_(L/(M+1))-R_((L+1)/M)))/((R_(L/M)-R_(L/(M+1)))(R_((L+1)/(M+1))-R_((L+1)/M)))=(C_((L+1)/M)C_((L+1)/(M+1))x)/(C_(L/(M+1))C_((L+2)/M))
(45)
((R_(L/M)-R_((L-1)/(M+1)))(R_((L+1)/M)-R_(L/(M+1))))/((R_(L/M)-R_((L+1)/M))(R_((L-1)/(M+1))-R_(L/(M+1))))=(C_(L/(M+1))C_((L+1)/(M+1))x)/(C_((L+1)/M)C_(L/(M+2))).
(46)

另請參閱

C-行列式, 經濟有理逼近, 弗羅貝尼烏斯三角形恆等式

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參考文獻

Baker, G. A. Jr. "The Theory and Application of The Pade Approximant Method." In Advances in Theoretical Physics, Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press, pp. 1-58, 1965.Baker, G. A. Jr. Essentials of Padé Approximants in Theoretical Physics. New York: Academic Press, pp. 27-38, 1975.Baker, G. A. Jr. and Graves-Morris, P. Padé Approximants. New York: Cambridge University Press, 1996.Brent, R. P.; Gustavson, F. G.; and Yun, D. Y. Y. "Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants." J. Algorithms 1, 259-295, 1980.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Padé Approximants." §5.12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 194-197, 1992.Weisstein, E. W. "Books about Padé Approximants." http://www.ericweisstein.com/encyclopedias/books/PadeApproximants.html.

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帕德近似

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韋斯坦因，埃裡克·W. "帕德近似。" 來自 —— 資源。 https://mathworld.tw/PadeApproximant.html

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