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Algebraic approximants to exp(z) and applications in construction of difference schemes of first order ODE
Applied Mathematics and Computation 149 (2004) 469–474 www.elsevier.com/locate/amc
Algebraic approximants to expðzÞ and applications in construction of difference schemes of first order ODE Feng Gao *, Ren-hong Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China
Abstract In this paper, an approximation to exponential function expðzÞ by algebraic functions is given. Its applications in construction of difference schemes of first order ODE are also shown. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Algebraic function; Algebraic approximant
1. Approximation by algebraic functions An algebraic function w ¼ uðzÞ of order ½n; m is defined by a polynomial F ðz; wÞ of degree n in z and of degree m in w as F ðz; uðzÞÞ 0;
F ðz; wÞ ¼
n X m X i¼0
aij zi wj
ð1Þ
j¼0
In the case of m ¼ 1, the algebraic function w ¼ uðzÞ defined by (1) is actually a rational function, since F ðz; wÞ ¼ a0 ðzÞ þ a1 ðzÞw 0 defines a function w ¼ a0 ðzÞ=a1 ðzÞ where a0 ðzÞ and a1 ðzÞ are polynomial of degree n. Consider the algebraic approximation to function f ðzÞ.
*
Corresponding author. E-mail address: [email protected] (F. Gao).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00154-1
470
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
Definition 1. An algebraic function w ¼ uðzÞ defined by (1) is called an algebraic approximant to f ðzÞ of order ½n; m if F ðz; f ðzÞÞ ¼ Oðzðmþ1Þðnþ1Þ1 Þ
ð2Þ
One can sees that in the case of m ¼ 1,the algebraic approximant to f ðzÞ is actually the Pade approximant to f ðzÞ denoted as ½n; nf ðzÞ . So algebraic approximant is a kind of generalization of Pade approximant. Shafer [1] once considered quadratic Pade approximants which is related to algebraic approximans to some extent.
2. Algebraic approximants to exp(z) The exponential function f ðzÞ ¼ expðzÞ satisfies functional equation f ða þ bÞ ¼ f ðaÞf ðbÞ and the weaker equation f ðzÞf ðzÞ ¼ 1
ð3Þ
So good approximants to expðzÞ should preserve the Eq. (3). This kind of function is said to be reversible. For an algebraic approximant w ¼ uðzÞ to expðzÞ defined by (1), it is reversible if the kth row of coefficients ðak0 ; ak1 ; . . . ; akm Þ is anti-reflective when k is even, and is reflective when k is odd. Take for example, the case k ¼ 0, m ¼ 1 then a00 ¼ a01 the case k ¼ 0, m ¼ 2 then a00 ¼ a02 ;
a01 ¼ 0
the case k ¼ 1, m ¼ 1 then a10 ¼ a11 the case k ¼ 1, m ¼ 2 then a10 ¼ a12 This is valid because when F ðz; wÞ ¼
n X m X i¼0
aij zi wj 0
j¼0
it is not difficult to see F ðz; w1 Þ ¼
n X m X i¼0
j¼0
i
j
aij ðzÞ ðw1 Þ 0
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
471
under the coefficient condition. We are now to get some algebraic approximants to expðzÞ. The Pade approximant ½1; 1ez is ð2 þ zÞ=ð2 zÞ So the algebraic approximant w ¼ uðzÞ to ez of order [1,1] is defined by ð2 zÞw þ ð2 zÞ 0
ð4Þ
One can sees that ð2; 2Þ is anti-reflective and ð1; 1Þ is reflective, so the function defined by (4) is reversible. The Pade approximant ½2; 2ez is ð12 þ 6z þ z2 Þ=ð12 6z þ z2 Þ So the algebraic approximant w ¼ uðzÞ to ez of order [2,1] is defined by ð12 6z þ z2 Þw þ ð12 6z z2 Þ 0
ð5Þ
It is also reversible. Let w ¼ uðzÞ is an reversible algebraic approximant to expðzÞ of order ½1; 2 defined by F ðz; wÞ ¼ ða00 þ a10 zÞ þ ða01 þ a11 zÞw þ ða02 þ a12 zÞw2 0
ð6Þ
where a00 ¼ a02 , a01 ¼ 0, a10 ¼ a12 . From (2), we have ða00 þ a10 zÞ þ a11 zez þ ða00 þ a10 zÞe2z ¼ Oðz5 Þ
ð7Þ
Expand ez and e2z into power series, substitute them into (7), compare the coefficients appearing in (7), we get a linear equation and one of its solution is a00 ¼ 3;
a10 ¼ 1;
a11 ¼ 4
ð8Þ
So the reversible algebraic approximant to expðzÞ of order ½1; 2 is defined by ð3 þ zÞ þ 4zw þ ð3 þ zÞw2 0
ð9Þ
Similarly, the reversible algebraic approximant to expðzÞ of order ½2; 2 is defined by ð15 þ 7z þ z2 Þ þ 16zw þ ð15 þ 7z z2 Þw2 0
ð10Þ
the reversible algebraic approximant to expðzÞ of order ½1; 3 is defined by ð11 þ 3zÞ þ ð27 þ 27zÞw þ ð27 þ 27zÞw2 þ ð11 þ 3zÞw3 0
ð11Þ
3. Applications in construction of difference schemes of first order ODE As is known, approximants to expðzÞ is of great value in constructing difference schemes of differential equations. For example, based on Pade approximants to expðzÞ, some linear symplectic difference schemes are made [2].
472
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
Now we show an important method to construct difference schemes of first order ODE by the use of the algebraic approximant to expðzÞ of order ½1; n which is obtained in Section 2. Given an initial value problem 0 y ¼ f ðx; yÞ yðx0 Þ ¼ y0 where f ðx; yÞ satisfies Lipschiz condition on ½a; b. Let a < x0 < x1 < x2 < < xk < < xN < b and h ¼ xkþ1 xk , k ¼ 1; 2; . . . ; N . we are now to construct its difference schemes. First, we can see yðx þ hÞ ¼ yðxÞ þ y 0 ðxÞh þ y 00 ðxÞh2 =2! þ y 000 ðxÞh3 =3! þ Define operator D as DyðxÞ ¼ y 0 ðxÞh, operator E as EyðxÞ ¼ yðx þ hÞ and operator I as IyðxÞ ¼ yðxÞ. Therefore EyðxÞ ¼ IyðxÞ þ DyðxÞ þ D2 yðxÞ=2! þ D3 yðxÞ=3! þ Then E ¼ I þ D þ D2 =2! þ D3 =3! þ ¼ eD
ð12Þ
Note (4) and (2), one can sees ð2 zÞez þ ð2 zÞ ¼ Oðz3 Þ
ð13Þ
When jzj is small enough, we have ð2 zÞez þ ð2 zÞ ¼ 0
ð14Þ
So when h is small enough, using operator D to replace z in (14), we get an operator equation ð2 DÞeD þ ð2 DÞ ¼ 0 thus ð2 DÞE ¼ 2 þ D Then ð2 DÞEyðxÞ ¼ ð2 þ DÞyðxÞ That is 2yðx þ hÞ hy 0 ðx þ hÞ ¼ 2yðxÞ þ hy 0 ðxÞ Using xkþ1 to replace x þ h, xk to replace x and yk to replace yðxk Þ, we get a difference scheme as follows 2ykþ1 hf ðxkþ1 ; ykþ1 Þ ¼ 2yk þ hf ðxk ; yk Þ
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
473
This is the famous Euler formula ykþ1 ¼ yk þ hðfk þ fkþ1 Þ=2 Second, from (9) and (7), we have ð3 þ zÞ þ 4zez þ ð3 þ zÞe2z ¼ Oðz5 Þ In the same way, when h is small enough, we use the operator equation ð3 þ DÞ þ 4DE ¼ ð3 DÞE2 to get ð3 þ DÞyðxÞ þ 4DEyðxÞ ¼ ð3 DÞE2 yðxÞ Note E2 yðxÞ ¼ yðx þ 2hÞ Then we can get a 4-stage 2-step difference scheme: ykþ2 ¼ yk þ hðfk þ 4fkþ1 þ fkþ2 Þ=3 which is known as Milne formula. At last, from (11) and (2), we have ð11 þ 3zÞ þ ð27 þ 27zÞez þ ð27 þ 27zÞe2z þ ð11 þ 3zÞe3z ¼ Oðz7 Þ When h is small enough, we can use the operator equation ð11 þ 3DÞ þ ð27 þ 27DÞE ¼ ð27 27DÞE2 þ ð11 3DÞE3 Note E3 yðxÞ ¼ yðx þ 3hÞ we can get a 6-stage 3-step difference scheme as follows: ykþ3 ¼ yk þ
27 h ðykþ1 ykþ2 Þ þ ð3fk þ 27fkþ1 þ 27fkþ2 þ 3fkþ3 Þ 11 11
Generally speaking, the reversible algebraic approximant to expðzÞ of order ½1; n will lead to 2n-stage n-step linear difference schemes of first order ODE.
4. Conclusion The algebraic approximant to a function is a kind of generalization of Pade approximant. The algebraic approximant to expðzÞ is of great value in constructing difference schemes of first order ODE.
474
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
Acknowledgements The paper was supported by the NKBRSF of China (G1998030600), the NNSF of China (10072013) and Higher Education Commission Doctoral Foundation of China (98014119).
References [1] R.E. Shafer, Quadratic pade approximants, SIAM J. Numer. Anal. (1972). [2] K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math. 4 (1986) 279–288.
Algebraic approximants to expðzÞ and applications in construction of difference schemes of first order ODE Feng Gao *, Ren-hong Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China
Abstract In this paper, an approximation to exponential function expðzÞ by algebraic functions is given. Its applications in construction of difference schemes of first order ODE are also shown. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Algebraic function; Algebraic approximant
1. Approximation by algebraic functions An algebraic function w ¼ uðzÞ of order ½n; m is defined by a polynomial F ðz; wÞ of degree n in z and of degree m in w as F ðz; uðzÞÞ 0;
F ðz; wÞ ¼
n X m X i¼0
aij zi wj
ð1Þ
j¼0
In the case of m ¼ 1, the algebraic function w ¼ uðzÞ defined by (1) is actually a rational function, since F ðz; wÞ ¼ a0 ðzÞ þ a1 ðzÞw 0 defines a function w ¼ a0 ðzÞ=a1 ðzÞ where a0 ðzÞ and a1 ðzÞ are polynomial of degree n. Consider the algebraic approximation to function f ðzÞ.
*
Corresponding author. E-mail address: [email protected] (F. Gao).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00154-1
470
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
Definition 1. An algebraic function w ¼ uðzÞ defined by (1) is called an algebraic approximant to f ðzÞ of order ½n; m if F ðz; f ðzÞÞ ¼ Oðzðmþ1Þðnþ1Þ1 Þ
ð2Þ
One can sees that in the case of m ¼ 1,the algebraic approximant to f ðzÞ is actually the Pade approximant to f ðzÞ denoted as ½n; nf ðzÞ . So algebraic approximant is a kind of generalization of Pade approximant. Shafer [1] once considered quadratic Pade approximants which is related to algebraic approximans to some extent.
2. Algebraic approximants to exp(z) The exponential function f ðzÞ ¼ expðzÞ satisfies functional equation f ða þ bÞ ¼ f ðaÞf ðbÞ and the weaker equation f ðzÞf ðzÞ ¼ 1
ð3Þ
So good approximants to expðzÞ should preserve the Eq. (3). This kind of function is said to be reversible. For an algebraic approximant w ¼ uðzÞ to expðzÞ defined by (1), it is reversible if the kth row of coefficients ðak0 ; ak1 ; . . . ; akm Þ is anti-reflective when k is even, and is reflective when k is odd. Take for example, the case k ¼ 0, m ¼ 1 then a00 ¼ a01 the case k ¼ 0, m ¼ 2 then a00 ¼ a02 ;
a01 ¼ 0
the case k ¼ 1, m ¼ 1 then a10 ¼ a11 the case k ¼ 1, m ¼ 2 then a10 ¼ a12 This is valid because when F ðz; wÞ ¼
n X m X i¼0
aij zi wj 0
j¼0
it is not difficult to see F ðz; w1 Þ ¼
n X m X i¼0
j¼0
i
j
aij ðzÞ ðw1 Þ 0
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
471
under the coefficient condition. We are now to get some algebraic approximants to expðzÞ. The Pade approximant ½1; 1ez is ð2 þ zÞ=ð2 zÞ So the algebraic approximant w ¼ uðzÞ to ez of order [1,1] is defined by ð2 zÞw þ ð2 zÞ 0
ð4Þ
One can sees that ð2; 2Þ is anti-reflective and ð1; 1Þ is reflective, so the function defined by (4) is reversible. The Pade approximant ½2; 2ez is ð12 þ 6z þ z2 Þ=ð12 6z þ z2 Þ So the algebraic approximant w ¼ uðzÞ to ez of order [2,1] is defined by ð12 6z þ z2 Þw þ ð12 6z z2 Þ 0
ð5Þ
It is also reversible. Let w ¼ uðzÞ is an reversible algebraic approximant to expðzÞ of order ½1; 2 defined by F ðz; wÞ ¼ ða00 þ a10 zÞ þ ða01 þ a11 zÞw þ ða02 þ a12 zÞw2 0
ð6Þ
where a00 ¼ a02 , a01 ¼ 0, a10 ¼ a12 . From (2), we have ða00 þ a10 zÞ þ a11 zez þ ða00 þ a10 zÞe2z ¼ Oðz5 Þ
ð7Þ
Expand ez and e2z into power series, substitute them into (7), compare the coefficients appearing in (7), we get a linear equation and one of its solution is a00 ¼ 3;
a10 ¼ 1;
a11 ¼ 4
ð8Þ
So the reversible algebraic approximant to expðzÞ of order ½1; 2 is defined by ð3 þ zÞ þ 4zw þ ð3 þ zÞw2 0
ð9Þ
Similarly, the reversible algebraic approximant to expðzÞ of order ½2; 2 is defined by ð15 þ 7z þ z2 Þ þ 16zw þ ð15 þ 7z z2 Þw2 0
ð10Þ
the reversible algebraic approximant to expðzÞ of order ½1; 3 is defined by ð11 þ 3zÞ þ ð27 þ 27zÞw þ ð27 þ 27zÞw2 þ ð11 þ 3zÞw3 0
ð11Þ
3. Applications in construction of difference schemes of first order ODE As is known, approximants to expðzÞ is of great value in constructing difference schemes of differential equations. For example, based on Pade approximants to expðzÞ, some linear symplectic difference schemes are made [2].
472
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
Now we show an important method to construct difference schemes of first order ODE by the use of the algebraic approximant to expðzÞ of order ½1; n which is obtained in Section 2. Given an initial value problem 0 y ¼ f ðx; yÞ yðx0 Þ ¼ y0 where f ðx; yÞ satisfies Lipschiz condition on ½a; b. Let a < x0 < x1 < x2 < < xk < < xN < b and h ¼ xkþ1 xk , k ¼ 1; 2; . . . ; N . we are now to construct its difference schemes. First, we can see yðx þ hÞ ¼ yðxÞ þ y 0 ðxÞh þ y 00 ðxÞh2 =2! þ y 000 ðxÞh3 =3! þ Define operator D as DyðxÞ ¼ y 0 ðxÞh, operator E as EyðxÞ ¼ yðx þ hÞ and operator I as IyðxÞ ¼ yðxÞ. Therefore EyðxÞ ¼ IyðxÞ þ DyðxÞ þ D2 yðxÞ=2! þ D3 yðxÞ=3! þ Then E ¼ I þ D þ D2 =2! þ D3 =3! þ ¼ eD
ð12Þ
Note (4) and (2), one can sees ð2 zÞez þ ð2 zÞ ¼ Oðz3 Þ
ð13Þ
When jzj is small enough, we have ð2 zÞez þ ð2 zÞ ¼ 0
ð14Þ
So when h is small enough, using operator D to replace z in (14), we get an operator equation ð2 DÞeD þ ð2 DÞ ¼ 0 thus ð2 DÞE ¼ 2 þ D Then ð2 DÞEyðxÞ ¼ ð2 þ DÞyðxÞ That is 2yðx þ hÞ hy 0 ðx þ hÞ ¼ 2yðxÞ þ hy 0 ðxÞ Using xkþ1 to replace x þ h, xk to replace x and yk to replace yðxk Þ, we get a difference scheme as follows 2ykþ1 hf ðxkþ1 ; ykþ1 Þ ¼ 2yk þ hf ðxk ; yk Þ
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
473
This is the famous Euler formula ykþ1 ¼ yk þ hðfk þ fkþ1 Þ=2 Second, from (9) and (7), we have ð3 þ zÞ þ 4zez þ ð3 þ zÞe2z ¼ Oðz5 Þ In the same way, when h is small enough, we use the operator equation ð3 þ DÞ þ 4DE ¼ ð3 DÞE2 to get ð3 þ DÞyðxÞ þ 4DEyðxÞ ¼ ð3 DÞE2 yðxÞ Note E2 yðxÞ ¼ yðx þ 2hÞ Then we can get a 4-stage 2-step difference scheme: ykþ2 ¼ yk þ hðfk þ 4fkþ1 þ fkþ2 Þ=3 which is known as Milne formula. At last, from (11) and (2), we have ð11 þ 3zÞ þ ð27 þ 27zÞez þ ð27 þ 27zÞe2z þ ð11 þ 3zÞe3z ¼ Oðz7 Þ When h is small enough, we can use the operator equation ð11 þ 3DÞ þ ð27 þ 27DÞE ¼ ð27 27DÞE2 þ ð11 3DÞE3 Note E3 yðxÞ ¼ yðx þ 3hÞ we can get a 6-stage 3-step difference scheme as follows: ykþ3 ¼ yk þ
27 h ðykþ1 ykþ2 Þ þ ð3fk þ 27fkþ1 þ 27fkþ2 þ 3fkþ3 Þ 11 11
Generally speaking, the reversible algebraic approximant to expðzÞ of order ½1; n will lead to 2n-stage n-step linear difference schemes of first order ODE.
4. Conclusion The algebraic approximant to a function is a kind of generalization of Pade approximant. The algebraic approximant to expðzÞ is of great value in constructing difference schemes of first order ODE.
474
F. Gao, R.-h. Wang / Appl. Math. Comput. 149 (2004) 469–474
Acknowledgements The paper was supported by the NKBRSF of China (G1998030600), the NNSF of China (10072013) and Higher Education Commission Doctoral Foundation of China (98014119).
References [1] R.E. Shafer, Quadratic pade approximants, SIAM J. Numer. Anal. (1972). [2] K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math. 4 (1986) 279–288.