Some Geodetic Applications of Clenshaw Summation

C.C. TSCHERNlNG - K. PODER 

* 

Some Geodetic Applications of Clenshaw 

Summation 

ESTRATTO DAL e BOLLETTINO D1 GEODESIA E SCIENZE AFFZNI D 

RIVISTA DELL' ISTITUTO GEOGRAFICO MILITARE 

ANNO XL1 - N. 4 - OTTOBRE - NOVEMBRE - DICEMBRE 1982

AKNO XL1 - BOLLETTINO D1 GEODESIX E SCIENZE AFFINI - N. 4, L982 

Some Geodetic Applications of Clenshaw 

Summation (*) 

C.C. TSCHERNING - I

1. - INTRODUCTION 

C.C. TSCHERNING - K. PODER 

It is well known, cf. e.g. Davis (1963, Theorem 10.1.1), that real orthonormal 

polynomials, P,(%), satisfy a three-term recurrence relationship, 

where a,(%) = a, .r + a,', and a,, a,' and bn are real numbers. 

Also other polynomials, or functions (real, complex or of several variables) 

fulfil such relations. Furthermore we frequently find recurrence relations between 

the polynomials and their derivatives or integrals. 

If P, depends on several variables, x = (x,, ..., xk), then both a, and b, in 

eq. (1) may depend (in a not necessarily linear manner) on one or more of the va- 

riables. 

In geodesy polynomials or functions, nhich fulfil recurrence relations are in 

frequent use. Most important are ordinary real or complex polynomials, Legendre 

polynomials, associated Legendre functions, solid spherical harnlonics, Chebychev 

polynomials and the trigonometric polynomials sin (WZ) , cos (m=), where z is real 

or complex and m an integer. 

We may take advantage of recurrence relations like eq. (l), when computing 

sums, derivatives 01- integrals of sums of series. This technique is called Clenshaw 

summation and it is described in section 2. The technique has been extensively 

used in physical geodesy for the computation of sums or derivatives of sums of 

Legendre series ( (Tscherning and Rapp, 1974), (Tscherning, 1976a)) and for the 

computation of double-sums of series of surface spherical harmonics (Gerstl, 1978). 

In section 3 we describe some new applications, namely for the summation of 

series of solid spherical harmonics and for the computation of integrals of (double) 

sums of surface spherical harmonics. 

Clenshaw summation may also be applied in the computation of sums of 

(complex) trigonometric polynomials, which frequently are used in geometrical geo- 

desy or in the evaluation of conformal mapprojections. Some of these applic a t' 

ions 

are described in section 4. 

The use of Clenshaw summation may result in considerable computational 

savings. However, even more important is the very favorable numerical properties 

of the method, see Clenshaw (1959), Smith (1967), Deuflhard (1976) and Gerstl

SOME GEODETIC APPLICATIOXS OF CLEKSHAW SUMMATION 35 1 

(1978). The fact, that the Clenshaw summation starts by summing the high order 

terms makes it especially well suited for most geodetic applications, because the 

higher order terms in general are small compared to the lower order terms. 

Notation : i, j, k, n, m will be integers, letters like x, y, p, s will be vectors 

with elements (e.g. X,, X,, ..., .zk), D! = aj/#, A, C, D, E will be matrices. 

2. - CLENSHAW SUNIMATION 

Let now P,(x) = P, be a general polynomial or function of one or more real 

or complex variables. For a series with real or complex coeficients y,,, we put 

where L! is the set of integration and 

The very simple idea behind Clenshaw summation is best illustrated (cf. (Deuflhard, 

1975)), if we express the evaluation of S on matrix form. Let us here for 

the sake of simplicity suppose, that for n = m + l we ha* bm+l = 0 in eq. 

(l), i.e. 

(The following is easily modified, if (5) does not hold). Then with


and from eq. (2), since A has determinant equal to 1 and hence is invertible, 

AT is an upper triangular matrix, for which the solution vector (A'r)-ly easily 

can be found. If we put sT = (S,, ..., sN), and 

then we have with SN+I = SN+Z = 0, 

and 

S = ~ r n . Pm(x). 

We get the first order derivative DiS by differentiating the matrix equation 

('7). However for the sake of simplicity we will now suppose that a,(%) is linear 

in xi and that b, is independent of xi, i.e. 

(The following is easily modified if this is not true). Then 

DiS = yT (Di(A-l) $0 $ A-' Di($o)) 

YT A-1 (- Di (A) A-' $0 + 

Di ($0)) 

= sT (- Di(A) A-' $0 + Di ($0)) .

SOME GEODETIC APPLICATIONS OF CLENSHAW SUMMATION 353 

Because in this case 

we have with 

and S;+, = S;+, = 0, 

and then 

1 

0 0 o.... 0 

l 

a,+, 0 0 

0' a 2 0 0 

It is easily seen, that the recurrence formulae for higher order derivatives, 

DDecomes 

This equation is slightly more general that eq. (17) of Cmith (1964). 

b 

Let us now suppose, that we for the integrals PI. = /a Pn(x) dx,, have a 

recurrence relationship of the following form, 

where C and E are matrices of the same simple form as the matrix A-I, so that

354 

and 

Then 

then 


= siT pio - gT (p3 (b) - (a)), 

Let us fin all^ regard a series of complex polynomials, Pn(z) = PR, $ i . PI,, 

where i is the imaginary unit, with coefficients yn -- yf, + i . yh. We must evalu- 

ate the real part of the sum Re(S) and the imaginary part of the sum Iln(S) inde- 

pendently, 

Let us put a,(z) = a: + i . a;, b, = b: $ i . bg in eq. (1) and S, = S: + i - S;. 

Then from eq. (8)

and 

SOME GEODETIC APPLICATIONS OF CLENSHAW SUMMATION 35 5 

Re (S) = S; . Re (P,,) - ~ h, . Im (Pm) 

Im(S) = S& . Im(P,,) + si . Re(Pm). 

3. - CLENSHAW SUMR'IATIOiV IN PHYSICAL GEODESY 

3.1. - EVALUATION OF (LOCAL) ISOTROPIC COVARIANCE FUNCTIOXS 

A (nearly) standard technique for the representation of local isotropic cora- 

riance functions uses a closed expression, F(r, Y', +), which is modified by a sum 

of a finite Legendre series, see (Tscherning Sr Rapp, 1974, section g), 

C (P, Q) = F (Y, Y', +) - 

P~(COS 

l+! , 

where P, Q are points in space, 4 is the spherical distance between the points, 

r, r' are the radial distances of P, respectively Q from the origin, on are constants 

and P, are the Legendre polynomials. With s = R"(Y~') and t = cos +, the sun1 

of the series can be expressed as 

We put p, (t, S) = sl'l P, (t) and we have 

i.e. an = - (212 - 1) st,'"r~ and b, = (1% - l)s2/~z. Eq. (8) and (9) can then be used 

for the evaluation of G(s, t) and eq. (12) and (13) for the evaluation of derivatives 

with respect to t. 

A normal potential of the Soniigliana-Pizzetti type (see Heiskanen & Moritz 

(1967, section 2-9)) may be expressed as the sum of a Legendre series plus the 

centrifugal potential. The sum and the derivatives of the sum can be calculated

356 C.C. TSCHERNING - K. PODER 

as discussed above in 3.1 using s = a/r and t = sin cp, where a is the semimajor 

axis of the reference ellipsoid and 9 is the geocentric latitude, see Tscherning (1976a). 

When computing gravimetric geoid undulations or deflections of the vertical 

using Stokes or Vening Meinesz integral formulae, the integration is in many cases 

only carried out over a spherical cap. In this case, the result may be improved 

in a least-squares sense by modifying the kernels by a function, 

N 

S (t) - CLo (2% + 1)/2 h, P, (t), (25) 

where h, are constants, see e.g. Jekeli (1980, eq. (73)). S(t) or its derivatives may 

easily be computed using a, = - (212 - l)t/n and b, = (n - l)/%. 

The external gravity potential of the earth, V, (or its corresponding anomalous 

potential, T) may be approximated by the sum of a series in solid spherical har- 

monics, 

G a 

V(?, A, Y) = C:=,U_, - %=, pm (sin 9) [COS 

( ; (mh) C," + sin(mA) S,"], (26) 

where A is the longitude, GJf the product of the gravitational constant and the 

mass ot the earth, P," the associated Legendre functions and C,", S," the coeffi- 

cients of the series. The conlputation of V ('2, A, r = a), i.e. for surface spherical 

harmonics, has been discussed by Gerstl (1978). We will here describe how Clen- 

shaw summation may be applied for solid spherical harmonic series and for the 

derivatives of the series. Note, that similar results easily may be obtained for 

internal solid spherical harmonics. 

It may of numerical reasons be of advantage to work with fully normalized 

or quasi-normalized associated Legendre functions. The latter will be denoted P,", 

with corresponding coefficients C:, S;, and we will here present results for these 

functions also. We now put q = air, t = sin cp, %L = cos cp and W = GM/r.

SOME GEODETIC APPLICATIONS OF CLENSHAW SUMMATION 

We first rearrange the double sum eq. (26), 

W 

V(+, h, r) = 

K XL, P;(t) [cos (mh) Cm + sin (ml) S:] qll , 

Y 

W 

Eq. (27) should be used when evaluating V in points with varying latitude, 

while eq. (28) should be used if is evaluated in many points with identical lati- 

tude and Y-value, because the sums 

then only need to be evaluated one time. (This idea is e.g. used in Rizos (1979)) 

The associated Legendre functions fulfil the well known recurrence relations 

(2n - 1) 12 + m - l 

it-m n-m 

F;-- t $ P;-:_, 

Correspondingly we have p; = P: . qn , p, = P: . qm , 

= 0, p:-, = 0 , 

The quasi-normalized Legendre functions are related to the un-normalized by 

P; = p; [ :-m)! l', pm = p; qn 

n + m)!

hence 


The values of a, and b, derived from these equations must then be used, when 

series with quasi-normalized coefficients are used. The reader may easily derive 

the corresponding equations for fully-normalized coefficients. For un-normalized 

coefficients I/:, may be evaluated using eq. (g), (9) with 

and 

The sum 

2n - 1 + m-1 q2,a;= cm .-2 

an = - tq, bn = *, urn - S:, 

n-m n - ~ 12 

V = C:=, W (V; cos (mh) + T'k sin (11th)) = C:=o W (S; cos (mh) + S; sin (mi,)) pm, 

can then be evaluated again using eq. (g), (9) with 

a, = - (21n - 1) z~q, b, = 0, v, = S; cos (W.) f S: sin (mh). (35) 

(Clenshaw summation degenerates here to the well-known Horner's schema for the 

computation of the value of a usual polynomial). 

However, when evaluating V, we inay take advantage of the recurrence rela- 

tions 

cos (mh) = 2 cos h . cos ((m - 1)k) - cos ((m - 2)h) (36) 

sin (mh) = 2 cos i, . sin ((m - 1)h) - sin ((m - 2)L) , (37)

ecause 

In this case 


pm cos (nzh) = (2wz - 1) .uq . $,, , . 2 cos A . cos ((m - 1)A) 

- (2m - 1) (2m - 3) ~t v2 $mpa . COS ((m - 2)A) 

$ sin (mh) = (2m - 1) uq . 9,-, . 2 cos h . sin ((m - 1)A) 

- (2% - 1) (2m - 3) uZ q2 $,M . sin ((m - 2)h) . 

a; = - (2% - 1) uq 2 cos h, b; = (2% - I) (2m - 3) uv2, 

and v; is given by eq. (33). 

However, it is in general more convenient to use eq. (36) and (37) for the 

direct computation of cos (m),) and sin (mh) and then eq. (35). 

Derivatives with respect to v are most easily computed using for 

The derivatives with respect to h are found by computing 1'; as in eq. (33), 

(34) and the using 

a - = C;=, m (V: COS (m),) - V; sin (7%~)) Q, , (42) 

The first and second derivative of V:, with respect to t and the second order deri- 

vative of Tf:, with respect to t and v may be computed using eq. (33) combined 

with eq. (12) c?i (13) iY- (40). The mixed derivative with respect to A and t may 

be computecl using eq. (12), (13) & (42), and the mixed derivative with respect

360 C.C. TSCHERXING - I 90. 

3.5. - COMPUTATION OF MEAN VALUES OF GEOID UNDULATIONS, GRAVITY ANOMA- 

LIES OR DEFLECTIONS OF THE VERTICAL 

Let the anomalous gravity potential, T, be given by a series expansion like 

eq. (26). However, the summation will start from TZ = 2. \Ye will here consider the 

computation of mean values over blocks where Q. q


with PI:-, = 0, PI: = to -t,, t, 7= sin (cp,), to = sin (qo), 11, = COS ((p,), U,, = COS ((P,,). 

Hence, from eq. (14) we get 

(n. + 

- 1) (n. - 2) 

c: = 0, a: = - (fi + l) (72 - ,m) 

which can be used for the first sum with respect to n. For the second sum, with 

respect to 772 we need 

e,, = 0,fn, = t /(~ + l), h - 0 . (474 

Hence, using eq. (16) - (18) tmo times, the integral of the series is computed. 

Mean geoid undulations are obtained using Bruns formula, i.e. the anomalous po- 

tential divided by the mean gravity. Mean free-air gravity anomalies are com- 

puted using coefficients multiplied by the factor - (7% - 1)lR. Mean values of de- 

flections of the vertical are computed using the fact, that they are (in linear ap- 

proximation) equal to the derivatives of T with respect to W and A, (multiplied 

by l/(ry) and l/(ryzs), respectively, where y is the reference gravity). 

4. - CLENSHAIIJ SURIMATION IN GEOMETRICAL GEODESY AND MAP- 

PROJECTIONS 

The fundamental work by Konig and Weise (1950) still forms the basis for 

the use of Clenshaw summation in connection with the ellipsoid and its mapping. 

This work, here referred to as KW, provides a wealth of series expansions, mo- 

stly in a standard form based upon the quantity (a + 11)/2, the half sum of the 

two semiaxes of the ellipsoid and n. = (a - b)/(a + h), the (( third flattening 1). It 

is therefore very easy to compute the actual coefficients at crun time)) from the 

actual ellipsoid used (instead of inserting numerical values in a programme or 

subroutine). 

'


A single problem not handled by KW is the computing of the Gaussian curvature 

of the ellipsoid as a function of Gauss-Kriiger (UTM) northings instead of 

the latitude (which not necessarily is available at run time). 

It is a remarkable fact, that the curvature needed in arc-to-chord corrections 

or scale corrections (see e.g. Bomford (1971, section 2.66) on transverse mercator 

projections), should not be computed at the latitude of the point, but at the latitude 

giving a meridian arc le~gth (from equator) equal to the northing on an ellipsoid 

scaled by the central scale of the projection - for UTM 0.9996. 

Therefore, one can find a series giving the Gaussian curvature of the form 

where T, are the Chebychef polynomials, 19 is a parametric northing varying from 

- 1 to + 1, when the actual northing varies e.g. from equator to the pole. The 

coefficients may be found at run time by numerical Chebychef analysis. As 

Clenshaw summation is applicable as discussecl in section 2. 

Kl4' gives all usual curvature radii and curvatures as trigonometric series in 

cosine of multiples of the double latitude. These formulae may be (and are) used 

if one wants to take advantage of the fact that the (( ellipsoidal )) terms are 2 -3 

magnitudes smaller than the main ((( spherical a) terms, but in many cases closed 

formulae based upon V = sq~t (1 + (e')Qos2B), where B is the geodetic latitude, 

are preferable. 

The length of the meridian arc as a function of latitude and its inverse function 

may be evaluated with 12 - 14 &git computing precision from series of the 

form 

G = B + C:=, p, sin (2kB) , (48) 

B = G + C:=, sin (2kG') (49) 

given in KWr page 51 and 53, where G is the meridian arc length in units of the 

meridian quadrant. 

These formulae extended to complex variables are the basis for a remarkably 

good set of formulae for transforming between geographical coordinates and trans- 

versal mercator coordmates as described in the next subsection.


The isometric latitude (defined by dq = dB/cos B for the sphere) and the lon- 

gitude are frequently used as the basis for map-projections because of its simple 

form, but problems arize the nearer the one approaches the poles. Obviously a kind 

of isometric longitude cos B dL might cure the problem. Considering the fact, that 

eq. (48) and (49) have no singularity at the poles and taking G and B as complex 

variables, 

G=N+~E and BZBr+iBi, (50) 

then we have for a sphere where 0, that B, = &' and B, = E . N and E 

become northing and easting. AT and E, when they are scaled by the meridian 

q~~acirant length. 

The formulae given in KPT7 for transformations between (ellipsoidal) geographical 

coordmates and transverse mercator coordinates are based upon the generalization 

of eq. (48) and (19) to complex variables, thus avoiding the singularity 

of the isometric latitude at the poles. IThen using summation to the fourth order, 

1 mm precision is obtained up to distances of 4000 km from the central meridian, 

and 1 - 2 mm from the pole. The undefined longitude at the pole is of course unavoidable. 

A parametric sphere (not to be confused ~ ~ith the osculating sphere - Schmiegungskugel 

in the Gauss-Schreiber approach) is used to ease the problems with 

the imaginary latitude on the ellipsoid. The full set of transformations both ways 

are two pairs each of a real trigonometric series and a complex one. Appendix 

2 lists a procedure (capable of transforming both ways) for this purpose and the 

initialization procedure for the constants of the transformation. 

5. - CONCLUSIONS 

We have here pointed out a number of applications of Clenshaw summation In 

geodesy. Its use gives many numerical as well as computational savings. 

However, other applications are possible, because the only condition is the 

existence of an equation like (1). Note, that such an equation may involve more 

terms, which only will make the algorithm as given by eq. (8) and (9) sljghtly 

more complicated. 

Acknoudedgement : It was Torben Krarup, who many years ago pointed out 

to us the many advantages of Clenshaw summation.


REFERENCES 

G. BOMPORD, Geodesy. Third Ed., Oxford at the Clarendon Press, 1971. 

C.W. CLENSHAW, Lq note on the summatio~ of Chebyshev sevies. MThC. Vol. 9, pp. 118-120, 

1955. 

P. J. DAVIS, Inte@olation and Afifivoximation. Blaisdell, Waltham, Mass., 1963. 

P. DEUFLHARD, On Algovithins f o ~ the sufnmalion of Certain Seecial Fz~nctions. Computing, 

Vol. 17, pp. 37-48, 1976. 

M. GERSTL, Ve7,gleich von A-llgo~&nen 311, Szinznzation v077 Kz~gelfliichenfunkiione~z. Veroff. d. 

Bayer. Komm. f.d. Int. Erdemssung, Heft ATr. 38, ~Miinchen, 1978. 

M. GERSTL, 0 7% the veczwsiue concputatiow of the integvals of the associafed Legendve f~lnctions. 

Manuscripta Geodaetica, Vol. 5, pp. 181-199, 1980. 

1V.A. HEISKANEX and H. XIo~r.rz, Physical Geodesy. Freemann, St. Francisco, 1967. 

C. JEKELI, Reducing fhe ewov of geoid undulation concpz~tafio.izs by modifying Stokes' function. 

Rep. Dep. Goedetic Science no. 301, Columbus, 1980. 

R. KONIG und K.H. \\'EISE, Mathematzsche GYzt?idlagen dev Hiiheveu Geodiisie und Kavfogvafihze. 

Erster Band. Sprmgel--Verlag, 1951. 

C. R~zos, An e-fficieilt co~z@utev iechvziqne for the evaluatiolz of geofiotential fvoin sphevical havmonics. 

Austr. J. Geod. Surv., So. 31, pp. 161-169, 1979. 

F. J. SMITH, A-ln algoritlm -fov sunzming orti~ogo~zal polynomial se14es and theiv derivalives with 

applicafions to cuwe;iitting and intei/polalion. Math. Comp., Vol. 21, pp. 629-638, 1967. 

C.C. TSCHERNING, Con~fiutation of seconrl-ovdev devivati~~es of the novmal poteiltial based 071 the 

repvesentatioiz a Legend9.e se7 les. Manuscripta Geodnetica, Vol. 1 pp. 71-92, 1976a. 

C.C. TSCHERNING, On the Chain-Rde Method fov Covnpztting Pofentinl Llevivatives. Nanuscripta 

Geodaetica, Vol. 1, pp. 125-141, 1976b. 

C.C. TSCHERNIXG and R.H. RAPP, Closed covaviance expvessions fov v~avity anovn,alies, geoid 

und~~dations, and deflections 0-f fhe vevtical implied by anomaly degvee va~iai~ce nzodels. Rep. 

Dep. Geodetic Science no. 208, Columbus, 1974.


appendix-l * page 1 17 08 81, 9.58; 

external real procedure apotdr(por C/ su, NI opderr G); 

value HI order; integer 14, order; array vo, G, C, sui 

comment 61 reg-no. 81013, programmed by C.r.Tscherningt july 1981. 

References: 

(1) Tscherniny, C.C.: On the Chain-rule method for Computing PO- 

- tential Derivdtives. 'Ianuscripte Geodaeticar Vol.lt 

- pp. 125-141, 1916. 

(2) This paper. 

The procedure computes the value and up to the second order derivatives 

of the potential of the Earth ( W ) or of i t s CorresPondiny 

anomalous ~~otentia! (T). 

The potential i s represented uy a series i n solid spherical her- 

monicst with un-nornalised or ouasl-normalized coefficfents. 

The ci~ain-rule i s used combined with the Clenshaw algorithm. 

The array C must hold the coefficients, C(O, U) L 1.0 for W and 

0.0 for T, C(1) = C(1, C(2) = C(?, l), C(3) = S(IP 1) etc. up 

to C((N+1)**2-l) = S(N* NI. 

Parameters: 

(a) CalL values: 

N The aDsolute va111e of 4 i s equal to the maximal desree and 

- order of the series. N negative indicatesr that the coeffi- 

- cients are cjiiasinori~~alized. 

order The maximal order of the derivatives (< 3 p.t.). 

PO Array holdirlg position information. Rounds (1:6). 

- 

oo(l) = n, the distdnce from the Z (rotation) axis, 

oo(2; = rn the distance from rhe oriqinr 

po(5)r po(4) cos and sin of the geocentric poLar angler 

P PO(;), ~o(6) sin and cos of the longitude. 

C Must oe declarei with hounds I-3: (N+1)**2-11 when the coef- 

- f l cients are un-norn~alised znd with bounds (-3: (N+3)**2-2) 

uiien the coeffCcients are qussf-normalized. C(1) to 

- C((li+l)**2-1) contains the coeiftciente sod we must have 

- C(-31 2 GM (the uroduct of the slass and the aravitatlonal 

constant, 

- 

C(-2) = a, the semi-major axis of the used ref. ellipsoidr 

C(-l) = the angular velocity (= 0, when dealing with T). 

C((N+I>**2+k) = sqrtck), k < 2*(N+1), when N < 0. 

(b) Qeturn values: 

6 

- 

- 

The result i s stored i n G as follows: 

G(Ot 1) = dGldxr G(U, 2) = dwldyr G(9r 3) = dwldzr 

G(?, 1) = dd14lddxr G(lr 2) = G(2, l! = dduldxdyr 

G(1, 3) = G(3, 1) = ddwldxdz, G(2r 2) ddWlddy~ 

G(2, 3) = G(3, 2) = ddW1dydz and G(3, 3) = ddilddz. 

'where W may he interchenaed witn T and the variables 

. 

X, y, z are tile Cartesian coordinates i n a LocaL (fixed) 

fra~ne with origin i n the poiit of cvaluationt x oosi- 

E 

- 

tivet North, y positive East and z positive i n the dlrection 

of the radius- vector^ (cf. ref.(?)! es. (4) and (5)ls 

The values of LJ or T rill be returned in gnotdr.


comment appendi x-l * oage 2 17 08 81r 9.58; 

cornmerit cont i~iued: 

(C) Call and return values: 

sm 

- 

- 8rray of dimension (fl;K*(N+I)-l), where K = case order of 

(2) 6, IU). Here are stored the partial sumsr cf. ref.(2)r eq. 

( 2 9 ) ~ of P(n, m)*(a/r)**(n+l-m)/p(m, m)*(C(nr m) or Stnr m) 

from rl=m to n=N, and the derivatives of tnese sums. This makes 

it unnecessary to recalculate these qdsntitier, i f the procedure 

is cdlled sut~sequently with the same value of t ana r, 

ar~d tne same oroer; 

comment ue use nohr that own booleans are initialised with the 

value false, first time tue ~rocedure is called in a proqrarn: 

if -r first then 

begin 

first:= true; oldt:= ?.Q; 

j:= abs(bJ): : I 1 i 2 2 i 3 : ~ 3*i; 1 4 : ~ 4*i; 

i5:= hi; ib:= 

end; 

6*i; i7:- 7*i; i8:x 8*i; i9:a 9*i: 

new:= abs(o1dr-r) > '-3 or abscoldt-t) > '-9 or oldord order 

or pole; 

i f ne* then 

beq i n 

oidr:= r; oldt:= t; oLdord:= order; 

end; 

quasi:= PI 0; i f qUdsi then N:= -11; nax:= (M+ 

S : - 2 / r ; sZ:= S*S; cmL(O):= 1.0; snil(D):= 

derivl:: order > U; deriv2:= order > 1; 

comment sml(m), cni(rn) dre the sln and cos of m* 

for m:= 1 steu 1 dntil ii do 

begin 

snilim) := smL(n1 )*cltcml(ml )*S[; 

cml(m):= cml(rnl)*cl-s~nl(ml)*si; ml:= m; 

end; 

1 ) **2 

u.0; 

(long 

NZ?:= 2*N+1; vm:= vxm:= vym:= vzm:= 0.3: sanrnl:= sqnnml:= 1.0; 

i f deriv2 then vxxm:= vyym:= vzzm:' vxva:= vxzrn:. vyzrn:= 0.0: 

km:= max; max:= i f quasi tnen Ntrnax else N; 

maxZ:= max+ri+l;


comment apoendix-l * oaqe 3 17 Oa 81, 9.58; 

comment we now use the Clenshaw algorithm, cf. ref.(2)r eq.(8 

- (13)r modified in an obvious way fqlloring ref.(l); 

for m:= N step -1 untii O do 

begin 

k:= km:= k~-(if T=O then 1 else 2); n2l!= N21; 

vs:= VC:= vsl:= vcl:= vxs1:= vxcl:= vzs:= vzc:= vzsl:= vzcl:= 

vxc:= vxs:= 0.0; 

if deriv2 then vxxc:= vxxs:= vxxcl:= vxxs1:s vzzc:= 

vzzs:= vzzcl:= vzzsI:= vxzc:= vxzs:= vxzcl:= vxzsl:= 0.0: 

cm:= cml(m); sm:= smlh); 

nnl := max-mt2; nl != lit?; npml := maxtmt2; 

i f deriv7 tnen sn2:= m*m; 

if new then 

for n:= N steu -1 until m do 

beain 

comment note that if -,quasi then nm2:rn-m+2r nml:=n-m+lr 

npml:= rltnltlr else the sametmax. Furthermore sqnpm2:= 

sqrt(ntm+Z)t etc.. Also note, that the values at the end of the 

LOOP are dsed later on (npml:= 2*m+lr sqnpml:= sart(2*mtl)r 

for examole): 

nm2:= nml; nml:= n,nl-l; npml := nnml-1; 

comment cf. ref.(2)t eo.(40); 

i f quasi then 

besi n 

comment cf. ref.(Z), eq.(30b); 

sqnm7:= sqnml; sqnml:= C(nrn1); 

sqnpmZ:= sqn~rnl; sqnpml:= C(npm1); 

sql := sqnn~l *sqnprnl; a1 := s*nZl /sql; 

hd:= -sl*sql/ (sqn1112*sqnpm2); 

end else 

heqin 

comment c 

a1 := s*n2 

e 11 d ; 

comment cf. ref .(2), 

vL:= vc1; vcl:= VC; 

v2:= vs?; vsl:: vs; 

if derivl tlien 

heqin 

ckz:= ck*nl; cklz 

comm~nt cf. ref.(Z 

v2:= vxc1; vxcl:= 

v2:= vxs1; vxs1:= 

v2:= vzc1; vzcl:= 

v2:= vzs1; vzs1:= 

ea. (53); 

= C k t k:= k-n21: 

, eq.(lOj; 

vxc; vxc:= vxcl*alttvcl*alu+v2*b2; 

vxs; vxs:= vxsl*al ttvsl*alu+v2*b2; 

vzc; vzc:= vzcl*alttv2*b2-ckz; 

vzs; vzs:= vzsl*alt+vZ*bZ-cklz; nl:= n;


comment auoendi X-l * page 4 17 08 87, 

i f deriv2 theri 

begin n2:= nt2; 

comment cf. ref.(2), eq.(41); 

v2:= vzzc1; vzscl:= vzzc; 

vzzc:= vzzcl*a7t+v2*b2+n2*ckz; 

vL:= vzrsl; YZZS~:= VZZS; 

vzzs:= vzzsl*alt+v2*b2tn2*cklz: 

comment cf. ref.(Z), eq.(12). The second order deriva- 

tive with respect to latitude; 

v2:= vxxcq; vxxc1:= vxxc; 

vxxc:= alt*(vxxcl-vcl)+2*alu*vxcltv2*b2; 

v2:= vxxs1; vxxsl:= vxxs; 

vxxs:= alt*(vxxsl-vsl)+2*alu*vxsl+v~*b2; 

erid; 

eornment cf. ref.(2), ea.(ln) end eq.(iO)r the derivative 

with respect to r and the latitude; 

v2:= vxrcl; vxzcl:= vxze; 

vxzc:= ~xzcl*al t+vzcl*alu+v2*bZ; 

v?:= vxrsl: vxzsl:~ vxzs; 

vxzs:= vxzsl*alt+vzsl*alu+v~*b2; 

end second order deriv.; 

end derivl; . 

end; 

If new then 

hegin 

su(rn);= VC; su(m+il):= vs: 

i f dcrivl then 

begin 

su(mti2):- vxc: su(m+i3):= vxs; 

su(mti4):= vzc; su(m+IS):= vzs; 

i f deriv2 then 

begin 

su(lnti6):: 

su(niti8):end; 

end; 

end else 

beqi n 

vzzc; 

vxzc; 

su(rnti7):;: vzzs; 

su(mti9) := vxzs; 

VC:= su(m); VS:= su(rnti1); 

if quasi theri 

begin 

sqnpml:= 

erid; 

C(mak2): sqnDrn2:= C(mex2+1); 

npml := nlax2; mak2:: max2-2; 

i f derivl then 

beq i n 

vxc:= su(n1ti2); vxs:= su(mti3); 

vzc:= su(mti4); vzs:= su(mti5); 

i f d ~ r i v ? then 

beqin 

vrzc:= su(mtl6); vzzs:= su(mti7); 

vxzc:= 

end; 

end; 

end; 

su(mti.5); vxzs:= su(rnti9);


appendix-2 * page 1 17 OU B 

References for all procedl~res are: 

(1) Pouer, Kwu anJ Frede NadSen: Status of the neu adjustment 

in Gree~land an3 tlle adjustment system of the Danish Geodetic 

Institute. Proceedings 2'Int. Sy:ncl. on Probl. Rel. Redefinition 

of North dmericdn Geodetic ldetworks, 

(2) This od~er. 

PD. 9-19/ 1978. 

(3) binsen Hansen, Mans (Ed.): ALGOL h. Users Menuall RCSL NO: 

31-D522, .ed, 197r. 

Fielu variahles dre used i n the brocedures. They are described 

in ref.(3), sectiorl 5.7. I n short# they act as pointers on 

elerents of an array, dhich then may he treated as booleanst 

inte~ers, lol>a (inteyers) reals or the first ekements of en array. 

................................. 

external lona procedure U-t-q 

- hi, E, Lit L, sat direct); 

value 

array sa; 

i geo hhen direct is true and the reverse otherwise. 

An alarm is produced when the check by the inverse transforrna- 

tion exceeds the toterance of 1.2 mm or an other vakue set by 

the user in sa(19) for utm->qeo or sa(2U) for aeo->utm. The 

alarm action uebends on the vdlue of tr-stetl~st a syetem verlable. 

tr-status >= 0 -> alarm exit, vrosrav termination 

tr-status, leftmost bit=l -> tr-status := tr-status + 1 

tr-status, 2 Leftm. bit=l -> report on diff. on cur r. outp. 

and tr-status := tr-status + 1 

The value of the procedure is the transformed tJ or 

in jeotype units, (see ref.(l)t p.15), 

L, t (call) lona 

The utm- or geoyranhical coordinetes input for transformation 

i n qeoty~e bnits. 

letltude 

At L (return) Lons 

The geoyraphical or utv-coordinates output from the procedure 

as transformed and checked coordinat~s in aeotvpe unlts. 

s a (call) arrey 

Transformation cor~stai~ts for direct and inverse transf. 

See fields in sa or set-utm-const for a descriatlon 

direct (cali) oooleen 

direct = t r u ~ => transformation utm, itn~, sb -> geogr. 

direct = false => transformation geogr -> utm, itm, 

external constarlts and procedures: 

c ten-sir?, clon-csin, t r -status:

C.C,. TSCHERNING - K. I'ODER 

comment appendix-2 *page2 170881,12.20; 

heqin 

Inteqer ir sr h; 

Long Rjr Lgr M-check, ~,checFi N-dlfr E-dif, 

- E-dcosr tol; 

real. Nor Ep, Rg-r, Lg-rt cos-BN, dN, dE: 

lony field 4-nr L@, E@, tol-f: 

array field hg-fr au-f. uta-f, ytu-f: 

hoolean utmr tod, neg-utm; 

ho6lean field utm-f; 

real urocedure arc-tan-h (X); 

value X; real X: 

arc-tanh := i f abs X < 0.95 then 

- (ln((1 t x)/(l - x?)lL.O? 

- else '300; 

comment fields i n sat see set-utm-const for details; 

Q-n 

E 0 

L 0 

b g - f 

qb-f 

atu-f 

utg-f 

to!-f 

- 

utm-f 

: = 

:= U-1) 

:= E0 

:= L0 

. - 

.- bq-f 

:= gb-f 

:= ytu-f 

:= uty-f 

t ( i f 

:= utg-f 

ii rect 

+ 28; 

comment transiormation seuuence; 

i := if direct then 1 else 3; 

h :c 4 - j; S := - i; 

comment check-va lues: 

utm := sa-utm-f; 

N-check := 14; €-check := E; 

tol := sa.tol-f; 

cohnNet\t transformation cases; 

 

 

s~h. NI E; 

Np := Np + clen-csin(sa.utg-f, 4, ?*Npr ~ * E D , dN, dk); 

Fp :c Fp + LIE; 

Comment sph. N, E = compl. soh. lat -> soh latr lna; 

cos-B14 := cos(Np?; 

Lg-r := arqtcos-RN, 91nh(Ep)); 

R := ara(cos-B!dr sin(Npl*cos (Lg-r) 1; 

comn,ent sph. lat, Lng -> ell. lat, lng; 

By-r := Ry-r t clen-sin(sa.gb-fr 4, 2*Rg-r); 

€g := Ry-rlrg; 

Lg := Lg-rlrg; L j := Lg + sa.LO; 

end case 1;


commer~t aunendi X-2 * oaqe 3 17 98 81) 12.ZU; 

comment case 2, trarlsf results: 

beqin 

U-t-q := 9 := 14 := By; 

- L := L := 1.g; 

end case 2: 

comment case 3, qeo -> utm) itm; 

beoin 

c* NE3 Nu NR: a,L refers to northinq and casting, 

- N,E refers to latitudr and longitude*> 

comment ell. [at) lng -> sph. tat, lng; 

neq-utm := I4 < O and utm; 

Rg-r := N*rq; 

Rg-r := Og-r + clen-sin(sa.bg-f, 4, Z*Bg-r); 

Ly-r := (E - sa.LO)*rg; 

comment sph. lat, Lng -> compl. sph. tat sph NI E; 

comment sph. normalized NI E -> ell. NJ E; 

Nv := Np + clen_csi~~(sa.qtu-f, 4, 2*Npr 

Ep := FP + dE; 

Rg := sa.U-ir*Nu; 

:= sa.U-n*Eu + sa.E0; 

:7 neg-utm then 

pg := + 20 ouo non ooo ouo; 

7*Epr d ~ dE); u 

end case 3; 

end transf cases; 

comment inlrev-dif for check; 

N-dif := Rg - N-check: 

E-di f := 

E-dcos := Ly - t-check; 

I f -I direct then E- COS := E-dcoS*cos(rg*N-check)i 

- 

comment error actions; 

if abs N-dif > to1 or abs E-dcos > to1 then 

begin 

comment output of N-check, E-check, N-dif, E-dif 

and the error-exit not shown here; 

tr-status := tr-status + 1; 

end error actions; 

end U-t-g; 

end;


external real procedure clen-csin 

(a, a, arg-r, arq-it R, I): 

value q, ary-r, arg-4; 

array a; inteaer g; real ary-rt arg-i, R, I; 

comment G1 r~g-no. YID14t pro~rammed by K.Poderr 1978. 

c ten-csin (call and return) real urocedure 

Comput~s the sun, of a series a(i)*sin(i*ara-r * i*i*arQ-l) 

(where 1 i s the in~~giriary unit) by Clenshaw summation 

from g down to 1. The coefficients are here real and 

the argument of sin i s complex. The real pert of the 

sum i s the value of the procedure. 

The procedure finds i n complex arithmetic an array h 

so that h(i) := a(?) + 2*cos(arg-r+j*ary,i)*h(i+?) - h(i+2) 

with li(y+l) = h(qt2) = U, see ref.(2), eq.(19) - (23). 

Then l(l)*sin(arq-r+j*arq-i) becomes the sum of the series. 

a (call) array 

The cuefficinent of the series to be summed deelerea 

as a(l:g) at least. 

Q (value call) integer 

The number of coefficients i n the series. 

arg-r, arg-i (vdlue call) real 

The real anu imaqinary vart of the argument of sine i n 

R, I (return) real 

The real and imdqinar) part of the sum; 

beq i n 

real r, it hr, hrl, hr2, hir hilt hi2r 

- sin-ary-r, cos-arq-rt sinh-ara-i, cosh-erg-i; 

integer t: 

 

sin-arq-r := sin(arq-r); cos-arg-r := cos(arg-r); 

sinh-arg-i 

r 

:= sinh(ary-i); cosh-arg-i 

. = 2*cos,drj,r*cosh-arg-i; 

:= cosh(ary,i): 

i := -2*si n-arg-r+sinh,a rq-i; 

 

hrl := hi1 := hi := U: 

hr := a(g); 

for t :- y - 1 step -1 until l do 

beain 

hr7 := hrl; hrl := hr: 

hi7 := hil; hi1 := hi; 

hr := -hr2 + r*hrl - i*hil + a(t): 

hi := -hi7 + ikhrl + r*hil; 

end; 

 

clerl-csin := 

R := r*hr - i*hi; 

I := r*hi + i*hr: 

end c len-csi n; 

end


external real procedure clen-sin 

- (ar j r 8rq); 

value J, arg; 

array a; 

inteqer g; 

real arrl; 

romment G1 reg-no. 81015, programmed by ~.Podert 1978. 

e len-sir (call and return) real procedure 

Computes the sum of a series of a(i)*sin(i*ary) by clenshaw 

sumnistion from g doin to l. 

The sum i s the value of the ~rocedure. 

The ~rocedure finds the elements of an array h 

so that h({) := a(i) t 2*cos(arq)*h(l+l) - h(i+Z) 

with h(gt1) = h(at2) = 0. 

Then h(l)*sin(arq) becomes the sum of the series. 

8 (csll) ilrray 

The coefficients of the series to be summed declared 

as a(l:y) at ledst. 

4 (value call) integer 

The number of coefficients i n the series. 

arq (value cdll) real 

The arqument of sine i n the series i n radians. 

hegin 

real hrr hrl, hr2t cos-ara; 

inteaer t; 

 

cos-arq := Z*cos(ary); 

 

hrl := d; 

hr := a(g); 

 

for t := g - 1 step -1 until 1 do 

beqi n 

 

hr2 := hrl: 

hrl := hr; 

 

br := -hrZ t cos-dry*hrl t e(t); 

end; 

 

elen-sin := hr*sin(arg); 

end clen-sin: 

end


apoendi X-; * paae b 17 08 81, '12.20; 

exterr~al integer brocedure set-utm-const(reg-laot sal; 

array reg-lab, sa; 

comment G1 reg.rio. 81910, programmed by Y.Poderr / nov. 1977. 

The procedure nroduces the constants needed I n the transformations 

bet~een transversal mercator and geoaraphical coordinates 

and the tolerances needed for then check of the transforqations. 

The transformation constants are qenerated from a 

req-label defininy a transversal mercator system. The formulae 

are taken frorn hO!dIG und WFISE : Mathematische Grundlaaen der 

hoheren Gendzsie und Kartoqr*phie, Erster Randr Berlin 1951. 

parameters 

set-utm-const (return) inteqer 

The lenyth of the array in pytes of the array needed for holding 

the11 constants. 

req-lab (call) array 

an array of 0 elements of which here are used the followina: 

req-lab(?) integer, here = i for i t m and = 4 for utmr 

rea-Lab(&) 

- 

Lonqr for utm zone number, 

of central qeridianr 

for i t m longitude 

reg-lah(6) geotypet seni-major axis of ellipsoidr 

req-lah(7) real, flattening/ 

req-lab(8) real, scale deviation from unity at central line, 

reo-lah(9) geotyne, ~astino parameter. 

S a (return) arrav 

The constants needeJ in the transformation. 

sa(1) = normalized neridian auadrant (lonot yeotype), 

sa(L) = pastinn at the central meridian (Lona, geotype), 

sac31 = Longitude of the central meridian (tonot qeotype), 

sa(4) - sa(7) = const for ell. geo -> sph. yeo (real) 

sa(8) - sa(l1) = const for sph. yeo -> ~ll. geo (real), 

sa(l2) - sa(15) = const for sph. 14, e -> ell. NI E (r@al), 

sa(l6) - sa(l9) = const for ell. 14, E -> sph. E (real), 

sa(2fl) = toler. for utm ii,put, 1.2'-10 mer quadr. (lonar yeotyoe), 

sa(21) = toler. for qeo input, 0.0fl0040 3x (loner yeotyve)r 

sa(22) = LooLeall for utmlitrnr true for utv (lonq), 

- utg uses only the Last byte as a boolean 

the user may cbdnje sa(Zn) - sa(22) for speciai checks. 

extrn. constants an3 procedures 

tr-status, rg (conversion factor from radians to d~yrees), 

conv-t-geo (which w i l l convert a real variable to seotype): 

heq i n 

inteoer r; real n, m; long W; boolean utm; 

array field bq-fr gb-f, gtu-fr utc-f; 

l onq field U-rr, EUL Ldr q-tol-f, u-tot-fr utm-f: 

comment fields i q sat oointiny at Individual variables or at 

first elelnents of arrays holding transformation constants; 

Q-n . - 4; 

E 0 . - U-I) + L; 

L 0 := EI) + L; 

bg-f := L0 ; soh. qeo *> 

qb-f := ba-f + 16; t* soh. geo -> ell. geo *> 

atu-f := gb-f + 16; ell. VtE *> 

utg-f := gtu-f t 16; soh. lea *> 

U-tot-f := uty-f t 20;


comment appenuix-L * nane 7 17 06 81, 12.2U; 

comment Lenyth of sa; 

set-utm-const := utm-f; 

comment test of coord system; 

r := r@g-Lab(2) extract 12: 

if r = 3 or r = 4 then 

beqin 

comment utmt lorig used as boolean; 

utm . .- - r = 4; 

sa.utm-f := if utni then (-1) else 0; 

Comment normalized meridian quadrant 

see K0niq und Weise p.50 (96)t p.19 (38b)r p.5 (2); 

n := req-lab(7)/(2 - rea-lab(7)); 

m := ~**2*(1/4 + n**2/64); 

U := (Lonq rep_lab(6))*(-n - req-lab(8) 

- + m*(l - rea,lab(8)))/(1 + n): 

sa.Q-n := Long reg-lab(6) + W; 

comment central eastinq and longitude; 

sa.EO := Long reg-lab(9); 

sa.Lo := if utrn the11 (((((lonj reg-lab(4) - 30)*2 - l) 

- shift 41)//15) shift 4) 

- else Lons req-Lab(4): 

cor,rnent chgck-to1 for transformstion: 

sa.v-tol-f := pi/2*1.Z1-1U*sa.q-n; 

9a.y-tot-f := conv-t-yeo(exteng 401 

- false add (1 shift 6 + 6)): 

comment coef of trig series; 

for r:=l step 1 untiL 4 do 

beg~n 

comment ell. yeo -> sph. geo., kW 0186 - 187 (51) - (52 

sa.bg,f(r):=case r of ( 

n *(-2 + n*(2/3 + n*(4/3 + n*(-82/45))) 

na*2*(5/3 + n*(-76/15 + n*(-73/91)), 

n**3*(-26/15 + n*34/21)~ 

comment sph. yeo - ell. qeo., KW p 

sa-yb-f(r)!=case r of ( 

n * ( Z + n*(-2/3 + n*(-2 

n**2*(7/3 + n*(-815 + n*(-22 

n**3*(56/15 + n*(-136) 1351, 

comment sph. NI E -> e l l 

sa.gtu-f(r):=cese r of ( 

n *(1/2 + n*(-2/3 

n**2*(13/48 + n*(-3/5 

n**3*(61/240 + n*(-1U3/ 

n**4*(4956l lI6l28U) 1: 

comment ell. ii, E -> sph 

sa.utg-f(r):=case r of ( 

n *(-l12 + n*(2/3 

n**Z*(-1/48 + n*(-1/15 

n**3*(-17/480 + n*37/d4UIr 

n**4*(-4397l161280)); 

end; 

end else ; 

end set-utm; 

end;

Some Geodetic Applications of Clenshaw Summation

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