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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
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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^ >,
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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
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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*
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RECURRENCE RELATIONS FOR CHEBYSHEV COEFFICIENTS 897
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)
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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)
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RECURRENCE RELATIONS FOR CHEBYSHEV COEFFICIENTS 899
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)
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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)-
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RECURRENCE RELATIONS FOR CHEBYSHEV COEFFICIENTS 901
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
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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\(y + 2* + 1)I> + * + 1)].fc = 0
r(i>), t//(i>) are the gamma and digamma functions, respectively.
5.1. Jvix) and dJvix)/dv. With suitable initial conditions, /„(x) is a solution
of the differential equation
x2y" + xy' + (x2 - v2)y = 0.
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RECURRENCE RELATIONS FOR CHEBYSHEV COEFFICIENTS 903
Writing Jvix) = iVx)vuvix), then
^ jvix) = (to/j^wsk) + ¿ uvix)} = jvixMm + &*r ¿ «„to-
Hence, on comparing this with (5.2),
f «VW = - Ê (- l)*04x)2*^(v + * + l)/[*!r(^ + fc + 1)].k=0
Writing u = uvix), w = buvix)/dv, then u and w, which are even functions
of x, satisfy the differential equations
xu" + (2i> + \)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.
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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.
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RECURRENCE RELATIONS FOR CHEBYSHEV COEFFICIENTS 905
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
1. M. ABRAMOWITZ &. I. A. STEGUN, Handbook of Mathematical Functions, Nat. Bur.
Standards, Appl. Math. Series No. 55, U. S. Government Printing Office, Washington, D. C,
1965.
2. C. W. CLENSHAW, "The numerical solution of linear differential equations in Cheby-
shev series," Proc. Cambridge Philos. Soc, v. 53, 1957, pp. 134-149.
3. C. W. CLENSHAW, "Chebyshev series for mathematical functions" in National Physi-
cal Laboratory Mathematical Tables, Vol. 5, Her Majesty's Stationery Office, London, 1962.
4. C. W. CLENSHAW & SUSAN M. PICKEN, "Chebyshev series for Bessel functions of
fractional order" in National Physical Laboratory Mathematical Tables, Vol. 8, Her Majesty's
Stationery Office, London, 1966.
5. L. FOX & I. B. PARKER, Chebyshev Polynomials in Numerical Analysis, Oxford
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.
7. Y. L. LUKE, The Special Functions and their Approximations, Vols. 1, 2, Academic
Press, New York, 1969.
8. Y. L. LUKE, Mathematical Functions and their Approximations, Academic Press, New
York, 1976.
9. Y. L. LUKE, Algorithms for the Computation of Mathematical Functions, Academic
Press, New York, 1978.
10. A. G. MORRIS & T. S. HORNER, "Chebyshev polynomials in the numerical solu-
tion of differential equations," Math. Comp., v. 31, 1977, pp. 881-891.
11. I. L. SCHONFELDER, "Chebyshev expansions for the error and related functions,"
Math. Comp., v. 32, 1978, pp. 1232-1240.
12. F. J. SMITH, "An algorithm for summing orthogonal polynomial series and their
derivatives, with applications to curve-fitting and interpolation," Math. Comp-, v. 19, 1965, pp.
33-36.
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