Recurrence Relations for the Coefficients in … Relations for the Coefficients in Chebyshev Series Solutions of Ordinary Differential Equations By T. S. Homer Abstract. Systematic

MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151

JULY 1980, PAGES 893-905

Recurrence Relations for the Coefficients

in Chebyshev Series Solutions

of Ordinary Differential Equations

By T. S. Homer

Abstract. Systematic methods are presented for obtaining recurrence relations for the

coefficients in Chebyshev series solutions of linear differential equations, of first to

fourth order, with polynomial coefficients. Polynomial approximations to certain ra-

tional functions are also discussed.

1. Introduction. In Morris and Horner [10], the Chebyshev series solution of

a linear fourth-order homogeneous differential equation was discussed in relation to

eigenvalue problems associated with simple boundary conditions. That investigation

provided a systematic method for obtaining the recurrence relation for the coefficients

in a Chebyshev series solution. The ideas are now applied to the solution of both

homogeneous and inhomogeneous equations of orders one to four. It is also shown

how the same approach can help in obtaining Chebyshev series expansions for certain

rational functions.

In the literature, there are many tables of Chebyshev expansions for mathemati-

cal functions, especially the elementary functions and the special functions. Among

such tables are those of Clenshaw [3], Clenshaw and Picken [4], Abramowitz and

Stegun [1], Luke [7], [8], [9] which all include references to further sources. Many

of the authors solve differential equations in order to find Chebyshev expansions, and

the methods are fairly standard. Nevertheless, the equations are often solved on an

ad hoc basis, and the aim here is to provide the data for quick and systematic con-

struction of recurrence relations for the Chebyshev coefficients. Indeed, the data can

be used to automate the solution of equations of appropriate type, but care should

be taken, for example, to investigate singular points of the equation, the convergence

of the solution, and the number of terms needed for a desired accuracy. In general,

any automatically generated recurrence relation should also be investigated analytically.

The equations to be solved are of the form

i dPy(1.1) £ P/j(*) TT = **). i= 1.2,3,4,0=0 dx

with suitable initial or boundary conditions.

Received June 25, 1979.

1980 Mathematics Subject Classification. Primary 65L05, 65D20.

© 1980 American Mathematical Society

0025-5718/80/0000-0115/$04.25

893

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

894 T. S. HORNER

/ is the order of the differential equation,

Ppix) is a polynomial, and for most of the problems will be quadratic,

and gix) is a continuous function, assumed to be expressed in Chebyshev series

form.

The Chebyshev polynomial of the first kind, Tnix), is used, where

(1.2) Tnix) = cos(« cos-1*).

The differential equation is solved on the interval -1 < x < 1, to give a series

as in Eq. (2.2) below. A change of variable can lead to solutions on other intervals

as will be seen in later sections.

2. The Method of Solution. The method of solution is the same as that in

Morris and Horner [10], and is stated briefly with regard to the solution of a second-

order equation.

Let / ^ L 2P2(X) = Cj + c2x + c3x ,

Pi(*) = c4 +c5x + c6x2,

P0(X) = C7 + CSX + c9*2 •

Thus, Eq. (1.1) becomes

(2.1) (Cl + C2X + C3*2) ~ + (C4 + C5X + C6*2) ¿ + (C7 + CSX + C9x2)y = *(*),

-1 <X< 1.

Let

(2.2) yix) = ¿' akTkix) = \a0 + «, 7» + «2r2(*) + • • • ,fc = 0 z

and let

(2-3) gix) = ¿' ^rfc(x).fc = 0

The derivatives ./^(x), r = 0, 1,2, are similarly given by,

(2-4) ?<'>(*) = E'^W.k=0

to include (2.2) when r = 0.

The following two results are used:

(2-5) C2xrTkix)=Z(f¡)Tk_p+2iix),

which is a generalization of the simple recurrence relation

2xTkix) = Tk_1ix) + Tk+1ix),

and

(2.6) 4-2,-4+1= 2*4^ >,


RECURRENCE RELATIONS FOR CHEBYSHEV COEFFICIENTS 895

which is a consequence of the fundamental theorem of calculus, and the integral

fTkix)dx = %{Tk+lix)Hk + 1) - (1 - 5kl)rfc_,(*)/(*: - 1)}.

Whenever a subscript is negative, the interpretation is

a<2=4'>.

The general method for solving (2.1) is then to substitute (2.3) and (2.4) to

obtain, after equating coefficients of Tkix), followed by repeated use of (2.5) and

(2.6), the recurrence relation

k+m I 9 ) k + m

E IZ'pftj {*- E »Í2.7*,,i=k-m (/=1 ) i=*-m I

(2-7) or f, * = 0,1,2,

c^* ),(*) = w7*>7*>

where W2fe) = (wjy°) is the 9 x 9 matrix in Table 1,

wjfc) is the /th column of w\k\

C = [Cj, C2, . . . , Cg J ,

a(fe) = [ak_m, . . . ,ak, . . . ,ak+m],

g = [%k-m> ■ ■ • >#k> • • • '8k + mi-

The maximum value of m is 4, but is related to the length of the vectors

a^ and g^k\ each of which has 2m + 1 components. Often in practice, m can

be taken to be less than 4, because some of the multipliers {wjp} and {cf} are

zero.

Once the general form of the recurrence relation is known, a suitable trunca-

tion point in the series (2.2), is chosen, (say at an), so that the series solution is

sufficiently well represented by the resulting polynomial. Then the appropriate

equations from (2.7), together with equations representing the initial or boundary

conditions, are solved for a0, ay, . . . , an. (See Clenshaw [2], [3], Fox and

Parker [5].) This is often done by solving an explicit set of algebraic equations

using standard methods such as Gaussian elimination, or iterative methods such as

successive overrelaxation. A common method involves back substitution in the re-

currence relation, followed by normalizing of the coefficients, using the initial or

boundary conditions. The actual method for solving the algebraic equations for

{ak} will not be considered further, but it is important that a stable, accurate

method is selected.

Differential equations of other orders can be solved in the same manner.

Thus, let Pßix) in Eq. (1.1) be written

pß(x) = E cc^ •oí=0

where d~ is the degree of Pßix) and will be usually taken as 2, but where there


896

• o—< III

il «o u

XX

o

C »—'

NI

X

X

»—'

oo

ca

H

I +

XM

X

+XX

CO

OIl J»! ii j<

Bei* ferf*



is allowance for higher degree polynomials. Now (1.1) can be written

<2-8> ¿ 2 c.^V» = gix),(3=0 a=0

where yiß) = dPy/dx*.

The recurrence relation from this equation is then found as

k + m i i dß )

L \Z Z c«,ßw<i-l,a,ß k(2.9)

i=k-m [ß=0 ct=0

k+m

= E w)-l,o,o^ * = o,i,i=k—m

or

crnf y*> = w<*> Vfc).

c is the vector with elements {ca^}, in the order of their occurrence in the

differential equation (2.8).

W\k^ = iw\kJß) is a matrix of 2m + 1 rows, and %ß=Q d~ (= y, say) columns,

these columns { wa »} being ordered to correspond to the ordering of the elements

of c. The 2m + 1 rows of w\k^ are numbered from -m to m, rather than from 1

to 2m + 1, to take account of certain quasi-symmetry properties of the matrix.

The value of m depends on the degrees of the polynomial coefficients {Pßix)},

and is given by m = max(a - ß + I). When all the coefficients are quadratic, then

m = / + 2, and S£=0 dß = y = 3(/ 4- 1), and hence the order of W,(k) is (2/ + 5)

x (3/ + 3).

The above notation is illustrated in Table 1, in relation to the second-order

differential equation with quadratic coefficients.

3. The Matrices { W¡k)}. Although it is desirable to have the matrices { w\k))

in explicit form, as in Table 1, it is impractical to exhibit such matrices for large /.

Thus with a modified notation for the elements of each matrix, namely writing

wia,ß = w(*> a' ß)> tne nonzer° elements of rows 0 torn, are given below. If the

value of any element of w\k^, (/ > 0) is fik), then the element found by reflection

about row 0, is -fi~k); and the entire matrix can be constructed.

The main tabulation below, corresponds to quadratic coefficients {po(x)}.

Sometimes, further elements of WJk^ are given at the foot of a table.

The elements are tabulated from the higher-order derivatives to the lower, but

with the polynomial coefficients being written in increasing powers of x. Except

for subsection 3.5, (/ = 0), the scaling of the elements is standardized so that there

are no fractional numerical elements when all the polynomial coefficients are qua-

dratics.

3.1. Nonzero Elements in the Lower Part of w\k) ■ (7 x 6).

wiO, 0, 1) = 8*

vv-(l,l,l) = 4(fc+l)


898 T. S. HORNER

w(0, 2, 1) = 4k

w(2, 2, l) = 2(* + 2)

w(l,0,0) = -4

w(0, 1,0) = 0

w(2, 1,0) = -2

w(l,2,0) = -l

w(3,2,0) = -l.

The above elements correspond to quadratic coefficients {p,(jc), p0(x)}. How-

ever, for the first-order equation all the elements of the matrix are given by simple

formulas. In fact,

wii,a,l) = ^(1/i(*_i)yk + i),

wQ, a, 0) =Vi(a -1 - 1) Wa-i+ \))\ '

where the combinatorial symbol (") is zero if r < 0, r > n, or r is noninteger.

3.2. Nonzero Elements in the Lower Part of W^f* • (9 x 9).

= I6kik- l)ik + 1)w(0, 0, 2

h<1,1,2

w<0, 2, 2

M.2, 2, 2

w(l,0, 1

w(0, 1, 1

w(2, 1, 1

H<1,2, 1

w(3, 2, 1

w(0, 0, 0

w(2, 0, 0

w(l,l,0

w(3,l,0

h<o, 2, o;

w(2, 2, 0

w(4, 2, 0

w(l, 3, 2

w(3, 3, 2

w(0, 4, 2

h<2, 4, 2

w(4, 4, 2

= 8(* - 1XA: + l)ik + 2)

= 8*(*2 - 3)

= 4(* - IX* + 2)(* + 3)

= -8(*- 1)(* + 1)

= Sk

= -4(fc- l)(/t + 2)

= -2ik-3)ik + 1)

= -2(*- l)(* + 3)

= -8fc

= 4ik - 1)

= -2(* + 1)

= 2ik - 1)

= -2*

= -2

= *- 1

= 6(fc + IX*2 + k - 4)= 2(A: - IX* + 3)(A: + 4)

= 6*(*2 - 5)

= 4(fc + 2X*2 + 2* - 5)

= (* - l)(it + 4X* + 5)



w(0, 3, 1) = 6*

w(2, 3, l) = -2(*-2)(* + 2)

w(4, 3, l) = -(*-l)(fc + 4).

3.3. Nonzero Elements in the Lower Part of W^k^ • (11 x 12).

w(0

wil

w(0

w<2

wil

w(0

w(2

w(l

w(3

vt<0

w(2

w(l

H<3

w(0

w(2

w(4

w(l

M<3

w(0

w(2

w(4

w(l

w(3

w(5

0, 3) = 32*(* - 2X* - 1)(* + 1)(* + 2)

1,3)= 16(* - 2)(* - IX* + 1)(* + 2X* + 3)

2, 3) = 16fc(* - 2)(fc + 2)(*2 - 7)

2, 3) = 8(* - 2X* - 1)(* + 2X* + 3)(* + 4)

0, 2) = - 16(fc - 2X* - 1)(* + IX* + 2)

l,2) = 32*(*-2X* + 2)

1,2) = -8(* - 2)(* - IX* + 2)(* + 3)

2, 2) = -4(ifc - 2)(* + IX*2 - 3* - 16)

2, 2) = -Mk - 2)(* - IX* + 3)(* + 4)

0, l) = -16/t(*-2)(* + 2)

0, 1) = 8(* - 2X* - 1)(* + 2)

1, \) = -4ik-2\k + IX*+ 5)

1, 1) = 4(*-2X*- IX*+ 3)

2, l) = -4*(*2 - 10)

2, l) = -12(*-2X* + 2)

2, 1) = 2(* - 2X* - 1)(* + 4)

0, 0) = 12(* - 2)(* + 1)

0, 0) = -40fc-2)(*- 1)

1,0) = -12*

l,0) = 4(/c-2X* + 2)

l,0) = -2(*-2X*-l)

2, 0) = 2(Jfc - 4X* + 1)

2, 0) = (* - 2X* + 5)

2,0) = -(*-2X*-l).

3.4. Nonzero Elements in the Lower Part of W\k^ • (13 x 15).

w(0, 0, 4) = 64*(* - 3X* - 2X* - 1)(* + IX* + 2X* + 3)

wil, 1,4) = 32(* - 3)(* - 2X* - IX* + IX* + 2X* + 3X* + 4)

HO, 2, 4) = 32*(* - 3X* - 2X* + 2X* + 3)(*2 - 13)w(2, 2, 4) = 16(* - 3)(* - 2X* - IX* + 2X* + 3)(* + 4X* + 5)

w(l, 0, 3) = - 32(* - 3X* - 2X* - IX* + IX* + 2X* + 3)


900 T. S. HORNER

w(0

w(2

w(l

w(3

w(0

w(2

wil

w(3

w(0

w(2

w(4

w(l

w(3

w(0

w(2

w(4

w(l

w(3

w(5

w(0

w(2

w(4

w(l

w(3

w(5

w(0

w(2

w(4

w(6

96*(* - 3X* - 2)ik + 2X* + 3)

- 16(fc - 3)(fc - 2\k - IX* + 2)(* + 3X* + 4)

-8(* - 3)ik - 2X* + IX* + 3X*2 - 5* - 32)-8(* - 3)(* - 2X* - 1)(* + 3X* + 4)(* + 5)

-32fc(fc - 3)(* - 2X* + 2)(fc + 3)

16(* - 3)(* - 2X* - IX* + 2)(* + 3)

-8(* - 3)(* - 2X* + 1)(* + 3X* + 8)

8(fc - 3X* - 2X* - IX* + 3)(* + 4)

-8*(*-3X* + 3)(*2 -22)

-8(fc - 3)(* - 2X* + 2)(5* + 19)

4(* - 3X* - 2)(* - IX* + 4)(* + 5)

24(fc - 3)(jfc - 2X* + 1)(* + 3)

-8(fc-3)(*-2X*- l)(* + 3)

-48*(*-3)(^ + 3)

8(it - 3X* - 2)(* + 2X* + 5)-4(* - 3)(* - 2X* - 1)(* + 4)

4(* - 7X* - 3X* + IX* + 4)

2(* - 3X* - 2)(* + 3X* + H)-2(*-3)(*-2X*- l)(* + 5)

24A:(A: - 3X* + 3)

- 16(fc - 3)(fc - 2X* + 2)

4ik - 3X* - 2X* - 1)

4ik - 3X* + IX* + 8)

-6(fc-3)(*-2X* + 3)

2(* - 3X* - 2)(* - 1)

4fc(*2 - 19)-(*- 17X*-3X* + 2)

-2(*-3)(*-2X* + 5)

(* - 3)(* - 2X* - 1).

3.5. The Matrix W^. If y is the rational function given by

!."o /E c«,ox°\y= E'gkT*(x), -i<x< 1,a=0 ) k=0

then the Chebyshev series expansion for y can be found from the recurrence relation

(2.9), with the nonzero elements of W^^ being given by

^'■^"¿(«(a-o)-



4. Numerical Evaluation of 2' akr'Tkix). The standard evaluation for the

finite sum

Ax)= L' "kTk(x)fc=0

is that of Clenshaw [3].

Let b

(4.1)

n+2 "n+1 u

and calculate

bk = ak +2xbk+1 -bk+2,

Then

fix) = Kib0 - b2).

k = n. 1,0.

This result is easily proved using

2xTkix) = Tk_lix) + Tk+lix).

Clenshaw, and Smith [12] show how the numerical values of f^r\x)/r\ can be

evaluated using similar schemes, and Hunter [6] points out how the method is re-

lated to synthetic division for evaluating a polynomial and its derivatives.

mthbk1+l = Vmk, * = 0, l,...,n,

\ttbrn+2_r = brn + i_r = (3

and calculate

b\ = 2*r+i + TxVk+i - bk+2, k = n-r,.

Then

(4.2) 1,0.

£^ = W>-b\),

for r = 0, . . . , n.

Then (4.2) incorporates (4.1), and the values of/and its derivatives can be

evaluated.

At the points x — 0, ± 1, the particular values for /, and its derivatives, are

fio)= e' (-iy+i«2*-2.ï=i

fiO)=Íi-iy+\2i-l)a2i_1,f=i

A0)=4É(-l/+1í2a2í,

(4.3)

i=l

/"'(0) = 41 (-l)'+1/(/ + 0(2/ + 1>2I+1.i=l


902 T. S. HORNER

/0)=E'«,>1=0

(4.4) <( /(l) = ¿' i\,i=l

AD = |Ê 0-i)/20 + iK-.1=2

/(-i)=E'(-i)''^ï=0

(4.5) J /(-!) = E(-l)'+1^,,i=i

A-D = lE (-îy'o-ix2o + ix-J i=2

For many simpie appUcations the values of a solution and its derivatives at

x = 0, ± 1 are sufficient for implementing initial and boundary conditions. How-

ever, the author has successfully used (4.1), (4.2) with the back-substitution method,

to impose conditions at any points.

5. Examples. As foreshadowed in the Introduction, the existence of the ma-

trices { W\k^} removes the need for a separate approach whenever a differential

equation with simple polynomial coefficients is solved in Chebyshev series. As ex-

amples, expansions are obtained for dJvix)¡dv, the derivative of the Bessel function

with regard to order, and for the integral ffi Jv(i) dt. Expansions for Jvix), /„(*)

are also obtained as steps in the calculations.

In order, these functions have series expansions:

(5.1) J„ix) = QÁxf ¿ (- l)k04x)2*/[*!r> + * + 1)],fc = 0

^Jvix) = Jvix)lni^x)

(5.2)- <ffl E (-i)fc0^)2k<K" + * + 0/[*!r(^ + * + i)],

fc=0

(5.3) T Jvit)dt = i1Áx)v+12 ¿ (- l)kiiÁx)2kl[k$y + 2* + 1)I> + * + 1)].fc = 0

r(i>), t//(i>) are the gamma and digamma functions, respectively.

5.1. Jvix) and dJvix)/dv. With suitable initial conditions, /„(x) is a solution

of the differential equation

x2y" + xy' + (x2 - v2)y = 0.



Writing Jvix) = iVx)vuvix), then

^ jvix) = (to/j^wsk) + ¿ uvix)} = jvixMm + &*r ¿ «„to-

Hence, on comparing this with (5.2),

f «VW = - Ê (- l)*04x)2*^(v + * + l)/[*!r(^ + fc + 1)].k=0

Writing u = uvix), w = buvix)/dv, then u and w, which are even functions

of x, satisfy the differential equations

xu" + (2i> + $u + xu = 0,

with i/(0) = l/riv + I), [u'iO) = 0], and

xw" + (2i> + l)w' + xw = -2m'

with w(0) - - iKv + l)/r(i> + 1), [w'iO) = 0].

An equation satisfied by the odd function «'(= du/dx) = v, is

xv" + 2iy -V l)u' + xv = -u

with [u(0) = 0], u'(0) = - V4/IX" + 2).

The series

«to = E' <**Wc), u(x) = ¿ 'ßkTkix/c),¡fc=0 k=0

CO

wto = E lkTkix/c), -c<x<c,k=0

are then found, after a change of variable, x —> ex, by solving the equations

xu" + i2v + iy + c2xu = 0, u(0) = l/r(i> + 1),

xv" + 2iv + l)v' + c2xv = -cu, v'iO) = -Vx/Yiv + 2),

xw" + (2v + l)w' + c2xw = -2ct>, w(0) = -}¡/(v + l)/r(i> + I),

where — 1 < x < 1.

The recurrence relations for {ak}, {ßk}, {yk} are

c2(* + l)aJt_3 + (* - l){4(Jfc + IX* + 2k - 1) - c2 }ak_!

+ (*+ 1){4(*- lX*-2c+ l)^c2}ak+1 +c2(*-l)«k+3 =0,

* = 3,5, ...,

c2(* + l)p\_3 + (* - 1){4(* + 1)(* + 2v) - c2 }ßk_l

+ (* + 1){4(* - IX* - 2») - c2 }ßk+i + c\k - l)ßk+3

= -2c(* + l)ak_2 + 4ckak - 2c(* - \)ak_2, * = 2, 4, ... ,

^2(* + l)7fc-3 + (* - 1){4(* + 1)(* + 2v - 1) - c2 }yk_l

+ (* + 1){4(* - IX* - 2v + 1) - c2 }yk+1 + c2ik - iyyk+3

= -4c(* + l)ßk_2 + 8ckßk - 4cik - l)ßk_2, k - 3, 5, ... ,

together with the appropriate initial conditions.


904 T. S. HORNER

In Abramowitz and Stegun [1], T(x), i//(x) are tabulated for x = 1(0.005)2

and x = 1(1)101. However, values of T(x), i//(x), for general x can be found by

using a Chebyshev series for 1/T(1 + x), 0 < x < 1, given by Clenshaw [3]. Not-

ing that

d 1 m + x)dx T(l +x) T(l +x)'

then Eqs. (4.1), (4.2), and the recurrence relations

T(x + 1) = xr(x), \pix + 1) = i//(x) + 1/x,

lead to the values of the gamma and digamma functions for any x for which the

functions are defined.

5.2. The Integral So JÁ{) dt Writing /* /„(r) dt = (&c)p+1z, then it follows

that the even function z satisfies

x2z" + (3v + 4)xz" + i2v + IX» + 2)z + x2z + (y + l)xz = 0,

2(0) = 2/1X1» + 2), z'(0) = 0, z"i0) = -l/[2iv + 3)riv + 2)].

The Chebyshev expansion

z(x)= £ ' akTkix/c),k=0

is found by letting x —* ex, and solving the equation

x2z'" + (3k + 4)xz" + i2v+lYy + 2)z' + c2x2z + (k + l)c2xz = 0,

z(0) = 2/T(k + 2), z'(0) = 0, z"(0) = -c2l[iv + 3)T(k + 2)], - 1< x < 1.

The recurrence relation is

c2ik + 1)(* + 2X* +P- 3)afc_4

+ 2(* - 2X* + 2){2(* + IX* + v - IX* + 2v - 2) - c2iv - 2)}ak_2

+ 2k{4ik - 2X* + 2X*2 -2v2 +V-1)- c2(*2 + 3v - l)}ak

+ 2(Jt + 2X* - 2){2(* - IX* - V + IX* - 2v + 2) + c2iv - 2)}ak+2

+ c2ik - IX* - 2X* - v 4- 3)ak+4 =0, * = 4, 6,... .

Conclusion. The matrices { W)k^} have been presented as a means of quickly

establishing the recurrence relations for the coefficients in the Chebyshev series solu-

tions of certain differential equations.

Second- and third-order equations were solved in the examples, and first- and

fourth-order equations can be solved in a similar way. The examples were of initial

value type, but boundary value conditions can be simply applied.

Linear differential equation eigenvalue problems can be formulated algebraically

as in [10], but shooting methods can be used successfully for both linear and non-

linear problems. A subsequent paper will report on these problems.



Special mention must be made of Miss Anne Cassidy who, as holder of the

Austin Keane Vacation Scholarship, checked and extended the matrices {Ff}*'} and

obtained many of the recurrence relations. Grateful thanks are also due to Miss

Kerrie Griffin who did her usual excellent job in typing the manuscript. Needless

to say, any errors and omissions are the fault and responsibility of the author.

Department of Mathematics

The University of Wollongong

Wollongong N.S.W. 2500, Australia

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Univ. Press, London and New York, 1968.

6. D. B. HUNTER, "Clenshaw's method for evaluating certain finite series," Comput. J.,

v. 13, 1970, pp. 378-381.

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