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Atomic Energy of Canada Limited
REMES 2: A FORTRAN PROGRAM TO CALCULATE
RATIONAL MINIMAX APPROXIMATIONS TO A GIVEN FUNCTION
by
J.H. JOHNSON and J.M. BLAIR
Chalk River Nuclear Laboratories
Chalk River, Ontario
May 1973 ^
AECL-4210
ATOMIC ENERGY OF CANADA LIMITED
REMES2: A FORTRAN PROGRAM TO CALCULATERATIONAL MINIMAX APPROXIMATIONS TO A GIVEN FUNCTION
by
J.H. Johnson § J.M. Blair
ABSTRACT
REMES2, a double precision FORTRAN program designed torun on a CDC 6000 ser ies computer, calculates rat ionalfunction minimax approximations (that is best approxi-mations in the sense that the maximum error in therange is as small as possible) to a given continuousfunction. The precision obtainable ranges from 1 to 24signif icant figures depending on the precision withwhich the function is specif ied. The program uses thesecond algorithm of Remes, combined with a s tar t ingalgorithm based on a suggestion of Werner. Variousoptions may be selected by input parameters, to generatee i ther single approximations or large segments of theWalsh array, and the resul t ing coefficients are auto-matically rounded off and tes ted for i l l -condi t ioning.
Chalk River Nuclear LaboratoriesChalk River, Ontario
May 19 73
AECL-4210
PvEMES 2: Programme FORTRAN permettantde calculer les approximations rat ionnelles minimax
pour une fonction donnée
par
J.H. Johnson & J.M. Blair
Résumé
REMES 2, est un programme FORTRAN a double
précision conçu pour fonctionner su" un calculateur
séquentiel CDC 6000, qui calcule les approximations
rationnelles minimax (ce sont les meilleures approxima-
tions que l'on puisse obtenir dans le sens où l'erreur
maximale, dans la gamme étudiée, est la plus petite
possible) pour une fonction continue donnée. La précision
pouvant être obtenue varie de 1 a 24 chiffres signifi-
catifs. Elle dépend de la précision avec laquelle la
fonction est spécifiée. Ce programme utilise le
deuxième algorithme de Remes, combiné avec un algorithme
de départ fondé sur une suggestion de Werner. Des
paramètres d'entrée permettent d'avoir recours à diverses
options pour engendrer soit des approximations simples,
soit de grands segments de la série de Walsh et les
coefficients qui en résultent sont automatiquement
arrondis et mis à l'essai pour détecter tout mauvais
conditionnement.
L'Energie Atomique du Canada, LimitéeLaboratoires Nucléaires de Chalk River
Chalk River, Ontario
Mai 1973AECL-4210
CONTENTS
Page
1. INTRODUCTION 1
2. THE NATURE OF MINIMAX APPROXIMATIONS 5
3. THE SECOND ALGORITHM OF REMES 6
4. STARTING VALUE ALGORITHM 8
5. THE ROUNDING STRATEGY 9
6. STRUCTURE OF REMES2 11
7. DESCRIPTION OF REMES2 SUBROUTINES 17
8. THE DATA STRUCTURE 18
9. THE I/O ROUTINES 21
10. SUGGESTED READING 22
11. REFERENCES 22
APPENDIX A: SAMPLE RUN OF REMES2 24
REMES2: A FORTRAN PROGRAM TO CALCULATERATIONAL MINIMAX APPROXIMATIONS TO A GIVEN FUNCTION
by
• .H. Johnson § J.M. Blair
1 . INTRODUCTION
The first question to be asked is why one should go to the
trouble of obtaining rational minimax approximations when
other simpler methods are usually available. The answer is
mainly efficiency; a rational function can represent any
calculation involving only arithmetic operations (that is, no
logical operations or loops) and the minimax approximation is
the most accurate of all the rational functions of a given
degree. Thus a function can be evaluated using a simple ratio
of polynomials to yield a pre-specified accuracy.
In order to understand why minimax approximations are the best
for most computer applications, we look at the "error curve"
f(x)-Pv(x)/Q
E(x) =w(x)
where f(x) is a continuous function of x to be approximated
and w(x) is a non-zero continuous weighting function. The
w(x) is introduced for convenience since w(x) = f(x) will
provide relative error and w(x) =. 1 will provide absolute
error. The function Rk£(x) = Pk(x)/Q£(x) represents any
rational function of degrees k over I.
We will now use R^ (x) to approximate f(x) in the interval
[a,b]. The accuracy of this approximation is the maximum
deviation observed in E(x). Let us denote this by
- 2 -
Mv. = max |E(x)i a a<x<b
To find the best R,.,(x), we must minimize M ^ . The resulting
function, to be denoted by R|,, is the minimax approximation.
To show how minimax approximations compare with other forms,the case of f(x) = exp x on [-1,1] will be considered. Fourdiffeaent approximations will be used:
A(x) = 1 + x + x2/2 + x3/6 + x4/24
B(x) = (12 + 6x + x2)/(12 - 6x + x2)
C(x) = .9996279 + .9979387x + .5028987x2
+ .1764862x3 + .0399629x4
D(x) = (12.2560247 + 6.1253843x + 1.OOOOOOOx2)/
(12.2560248 - 6.1253844x + 1.OOOOOOOx2)
4(A(x) is the Taylor series expansion truncated to terms in x ;
B(x) is the Pade approximation (first four derivatives agree
with exp x at x=0). C(x) and D(x) are the corresponding mini-
max approximations R-J n and R£ 0 using the relative error
criterion.)
Consider the maximum relative errors of each of these approxi-
mations:
MA = 1,94 x io'2
MB = 1.47 x 10"3
Mc = 5.03 x 10'4
MD = 8.68 x 10"5
It is easy to see that the minimax approximations are much
better., and also that Dfx) is almost a whole power of 10
better than C(x), suggesting that this is the best approxi-
mation to use. This is all shown quite clearly in Figure 1
where the error curves for each approximation are displayed.
- 3 -
3filltn3cj
- 4 -
The plot of error curves also demonstrates other interesting
phenomena. The Taylor series and Pade approximations are
very accurate close to the origin x=0 and very quickly deteri-
orate as one moves away from this point. On the other hand,
the minimax approximations are uniformly accurate over the
whole interval and show a characteristic alternating behaviour.
The error curves resemble the Chebyshev polynomials of appro-
priate degree CT (x) = cos(arccos(nx)); thus one way of
obtaining near-minimax polynomial approximations is to trans-
form the Taylor series to an expansion in terms of Chebyshev
polynomials and truncate it after the appropriate number of
terms. If the coefficients decrease sufficiently rapidly as
n gets larger, the absolute error is almost all due to the
first omitted term, a Chebyshev polynomial.
The following technique is suggested when all algorithmic
methods for a particular f(x) are too slow or take up too much
core memory. If a function is to be evaluated very many times
the improvement by this method will be quite significant.
First, a function subroutine is written to give the desired
function to about four more digits of accuracy than the approx-
imations are to have. The header card for this routine must be
DOUBLE PRECISION FUNCTION FN(X)
where X is a double precision variable. Thus, if 10 digits
of precision are desired in the approximations, FN is made
accurate to at least 14 or 15 digits of precision. Note that
there must be no discontinuities in FN(X) and that the closer
FN is to being analytic the better the approximations will be.
The function FN may not be the desired function itself but a
related function from which the desired function may be cal-
culated. In this case w(x) would be adjusted to provide
reasonable error properties in the evaluation of the desired
function rather than FN. If a non-standard weighting is desired,
it can be supplied by the double precision function WN(X).
- 5 -
Now, assuming a suitable FN has been written, it is loaded
with the REMES2 program and tested with various interval sizes.
The splitting of an interval improves the accuracy over each
sub-interval at the expense of a more complex approximating
function. It is best to use the largest intervals possible.
The interval [a,°°) may be transformed to [0,1/a] by approxi-
mating f(1/x).
When the auxiliary function FN and the sub-interval splitting
have been determined, the rest is done by REMES2. A control
deck specifying what approximations to calculate and display
is made up and the REMES2 program is run to obtain the results.
A suitable approximation may be extracted from the output and
be incorporated into any program requiring that function.
2. THE NATURE OF MINIMAX APPROXIMATIONS
The example of Figure 1 demonstrates several features of error
curves of minimax approximations. In the case of both C(x) =
R| 0(x) and D(x) = R?j 2(x)> i1: is found that the extrema
alternate in sign with equal amplitude six times. In general,
it can be proven that the error curve will alternate this way
not less than k+Jl+2-d times. The degeneracy d is the
order of the largest common factor of the numerator and
denominator of the true best approximation. True degeneracy
occurs when an even or odd function is approximated on an
interval symmetric about zero, when a periodic function is
approximated on more than one period, and in some other path-
ological cases. It can be avoided for even functions by using
only the positive half of the interval; odd functions may be
divided by x to provide even functions on which the above
strategy may be applied. For periodic functions one need only
make sure that not more than one period is used for an interval,
- 6 -
Thus, true degeneracy can usually be easily detected and
remedied. However, as with other numerical methods using
finite precision numbers, there is a fuzzy region in which
an answer is known to exist but for which the methods pro-
vided are unusable. For the second algorithm of Remes this
phenomenon, termed near-degeneracy, occurs when numerator
and denominator have zeros that are very close together,
usually near one of the end points of the interval. Cal-
culations are very difficult unless accurate starting values
are given. When REMHS2 detects problems with convergence,
it assumes near-degeneracy and works around it, simply avoiding
the difficulty.
For a more thorough discussion of minimax approximations,
Ralston (1967) or Hart (1968) are quite readable. The paper
by Cody (1970) is quite interesting since he has developed a
method for finding near-degenerate approximations. Cody's
method of artificial poles has not been included in REMES2
although it may be at some later date.
3. THE SECOND ALGORITHM OF REMES
The algorithm used by REMES2 is usually referred to as the
second algorithm of Remes. It is an iterative method that
uses an initial guess for the true minimax approximation and
from it calculates a better guess. In general this necessitates
the availability of good starting values (see "Starting Value
Algorithm").
The algorithm is based on the theory of minimax approximations
that states that the error curve alternates at least k+&+2-d
times. It, in effect, forces the alternation property onto
the error curve by making it pass successively through
(xQ,X), (X;L,-X), ..., (xk+J,+1, (-l)k+!l+1X) where X is to be
determined. More specifically, the non-linear system of
equations
- 7 -
E(xi) =
kZJ=0
1 + I
i = 0,1, .. .,k+«. + l are solved for a^a-j^ ak'bl'b2 ''" * 'b&
and X. The a's and b's are just the coefficients of a
rational function approximating f(x). The equations can be
solved by guessing a value for X and using the first k+«,+l
equations to solve for the a's and b's; the a's, b's and
guessed X are substituted into the k+A+2 equation which
now is effectively a function of X. The root of this equa-
tion may be determined using the secant method. For details
of this method, see Ralston (1967). If something goes wrong
during the Remes iteration, an attempt is made to restart
the algorithm with the sign of X changed. The user also has
the option of selecting the first rather than the (k+il+2)th
equation to calculate X (see the description of the B module).
The effects of these features are not known although they made
convergence possible in one or two cases.
The true extrema of this error curve provide better approximations
to the actual minimax values. In a neighbourhood of each of
the (x.,(-l)1X) there is an extremum of the same sign that
becomes the new approximation to x,. The search for maxima is basei
on the method used by Ralston (1967), but is modified to pro-
vide better convergence in some pathological cases.
The algorithm terminates when the error curve has been
sufficiently levelled (extrema have the same magnitude).
This occurs when X, which must be less than each extremum
in magnitude, agrees with M (the maximum error) sufficiently
well.
- 8 -
A more complete discussion of the algorithm may be found in
Ralston [1967); the discussion and flowchart there form the
basis for REMES2 although the starting algorithm was not
implemented.
4. STARTING VALUb ALGORITHM
The importance of good starting values for the Remes algorithm
should not be underestimated. In cases of near-degeneracy
the starting values must, be quite accurate for the method to
converge at all. A method that seems to work quite well is
one suggested by Werner (.1968) . The idea is to use the critical
points ot known nearby approximations to predict starting values
for the case under consideration. A nearby approximation is
chosen and the critical points are transformed by arccos
(. (2x- - [a+b))/ (b-a)) . The required number of points are forced
in using linear interpolation on the transformed points. The
reciprocal transformation is used to re-scale the points to the
interval [a,b]. The advantage of this approach is that extrema
of Chebyshev polynomials are transformed onto extrema of
Chebyshev polynominals of different degrees. Werner's sugges-
tion involves a similar transformation using zeros rather than
maxima. The effect of the modification is not precisely known.
RHNiliS2 checks neighbouring approximations according to the
strategy shown in Figure 2. For the purposes of this report
all Walsh tables are indicated with horizontal rows representing
constant k+X, and vertical rows constant k-Z. The program takes
the first case that has been approximated successfully and uses
it. If none are within the search area, the maxima of the
Chebyshev polynomial of appropriate degree are used. It is
recommended that lower order approximations are done before
high order ones, however, since this way the effectiveness of
the starting algorithm is the greatest. For this purpose the
command I 2 -2 5 is useful. If information from the WALSH table
. 9 _
suggests that a particular approximation is more suitable
than the one usually chosen, the A command may be used to
select the starting approximation.
Figure 2: Search Strategy for Starting Value Algorithim
Approximation for which starting values are desired.
(Numeral indicates order in which each relative position is
tested.)
The A module allows the user to specify a linear combination
of known approximations. It generates starting values using
each of the suggested known approximations and takes the
specified weighted average. For details see the discussion
of the A module.
5. THE ROUNDING STRATEGY
For the convenience of the user of REMI3S2, coefficients of
desired approximations may be rounded to a suitable number
of decimal places and displayed. The format and the rounding
strategy are those suggested in Hart (1968). For the power
polynomial representation, the quantities
- 10 -
g = min | Q ( x ) | , £ = max | f ( x ) | and min | ^ |m i n a<x<b m a x a<x<b a<x<b x1
are ca l cu l a t ed on a gr id of 129 equal ly spaced p o i n t s . For
the polynomial case
. a | < iiLfM _ n i n ,1 1 k+1 x x 1
is used and for the rational case
.a . | < m i n|1 1 1 + F k+1 xmax
where a^jb- are rounded coefficients.
Coefficients are rounded to n places beyond the decimal point,
where n satisfies
0.5 * 10"n < b < 0.5 * 10" n + 1
where b is the bound determined above. When the Chebyshev
form is used, x 1 must be replaced by T. (x) and x° by JgT (x)
in the above bounds calculations.
See the discussion of the K and L modules.
- 11 -
6. STRUCTURE OF REMES2
The program is set up as a number of modules that may be
called into operation by data cards. A typical run involves
reading a data card, interpreting the parameters, and trans-
ferring to the specified module. When the module has completed
its alotted task, it transfers back to the input section. This
continues until the data is exhausted. To understand REMES2,
one must know the data card format and the functions of each
module.
Each data card initiates a particular module. For identifica-
tion purposes, each module is assigned a letter that is punched
in column one of the data card. Since most of the modules
require additional information, a number of parameters may be
given in columns 2 to 80. The rules are simple; the parameters
must be integers with or without sign and must not contain em-
bedded characters. Unspecified parameters are set to zero. An
unnecessary parameter may be omitted only if it is at the end
since the integers are put into correspondence with parameters
as they are found on the card. The meanings of the parameters
can be fcund under the appropriate sub-headings below. A
complete oxample of the use of REMES2 along with input is
given in Appendix A.
The N module sets a flag to indicate that no RDAT file has
been supplied. This means that a new file must be created
before it can be opened. If required, the N card must be
the first card in the control deck. The operating system
will print an error message and terminate the run if this is
not used correctly.
The B module causes various parameters to be defined for a
run. This card must occur before any A, I, K, L, X or P cards
and stays in effect until the next B card. If the case to be
- 12 -
run is a new one, the information must be provided on data
cards. The parameters for a B card are (1) a case number (an
integer between 1 and 999) and (2) the number of data cards to
be read in by the B module. The first card contains the bounds
for the approximation in (2D40.40) format. Following data
cards contain a message of up to 240 characters to be inserted
in the data set header. (3) The error criterion to be used.
Zero means absolute error, one relative error, and two means
the function WN will be used as a weight function. (4) A
display flag. If not zero, certain intermediate calculations
are printed. (5) If not zero, the rational functions are cal-
culated in terms of Chebyshev polynomials. If zero, power
polynomials are used. (6) If not zero, the first equation
(instead of the last) is used as a residual function by RIPPLE
to obtain the new estimate of the amplitude. (7) If not zero,
this specifies a data set from which starting values may be
directly obtained. This is useful for forcing recomputation
of a data set since the starting values will be the true critical
points. In particular a data set may be transformed from power
form to Chebyshev form by creating a new data set with parameter
5 set to a non-zero value and parameter 7 set to the power form
data set number. Particular approximations are transformed by
I and A cards. The information from parameters (1), (3), (5),
and the message and bounds are stored in the data set header on
the RDAT file and need not be supplied on subsequent runs. If
they are supplied, they will be ignored; if a non-zero length
message is supplied, it will replace that on RDAT.
The A module enables the user to calculate a particular approx-
imation. The first two parameters indicate the numerator and
denominator degrees of the desired approximation; the remaining
parameters are interpreted in groups of three. The first two
of each group indicate a known approximation that is to be used
to generate starting values by the interpolation formula dis-
cussed under Starting Value Algorithm. The third of each
- 13 -
group is a weight to be used in the weighted average that is
taken of the starting values provided; if zero, this number
is converted to 1. The weighted average is used to start the
iteration.
By use of these weights various types of Newton interpolation
may be done; see Figure 3 for examples of useful Newton weights.
It is not yet known how useful this weighted average technique
is in practice although it is hoped that a judicious use of
this feature in conjunction with the P module should allow
many difficult approximations to be obtained.
It may be useful to feed in starting values from cards. This
may be done by setting parameter 3 to a negative number. The
magnitude of the number specifies the number of cards to be
read in. The numbers must be one to a card in D40.40 format
and must be in increasing order. The two endpoints must not
be included since they are added by the A module. Rescaling
by interpolation is done unless parameter 3 is equal to the sum
of the first two in absolute magnitude.
If parameter 4 is negative, the next data card contains an
estimate of the negative logarithm of the amplitude of the
error curve (the number printed by WALSH).
If parameters 3, 4 and 5 are all zero, the nearest neighbour
is found and used to obtain starting values.
- 14 -
Figure 3: Some Useful Low Order Newton Interpolatioa Weights
Linear (where A, B, and C equally spaced)
A A = 2B - C
B 2B = A + C
C C = -A + 2B
Quadratic
A A = 3B - 3C + D
B 3B = A + 3C - D
C 3C = -A + 3B + D
D D = A - 3B + 3C
Cubic
A A = 4B - 6C + 4D - E
B 4B = A + 6C - 4D + E
C 6C = -A + 4B + 4D - E
D 4D = A - 4B + 6C + E
E E = -A + 4B - 6C + 4D
e.g. A 7 5 8 4 1 6 6 3 5 7 - l i s a possible quadratic inter-
polation form to obtain starting values for Ry 5(x).
The I module allows one to cause a number of approximations to
be calculated without setting up control cards for each. The
first two parameters indicate the first approximation to be
tried. The third parameter indicates the width (or depth,
depending on the fourth parameter) of the band and the fifth
the number of rows (or columns). If zero, the value 100 is
used. The fourth parameter indicates which of three directions
the band is to proceed in. If one or zero, the band proceeds
downward with rows going to the left. In this case the third
parameter gives the width and the fifth parameter the depth of
the band (see the example in Appendix A). If it is two, the
band goes to the left and if three, to the right. In both of
- 15 -
these cases the third parameter is the depth and the fifth
parameter, the width of the band.
An example may be useful. The card I 2 -2 5 is equivalent
to the following A commands:
A 2-2
A 1 -1
A 0 0
A -1 1
A -2 2
A 2 -1
A 1 0
A 0 1
A -1 2
and so on. It generates the entries of the Walsh array in
the order indicated by the arrows in Figure 4-
Figure 4: Approximations Done by I 2 -2 5 Card.
- 16 -
If a particular approximation is impossible or has encountered
numerical problems, a counter is incremented to tally the
number of consecutive failures. If two full rows of failures
occur, the I module exits. Any success clears this countei .
An advantage of this approach is that a data card does not
have to be repunched if a run terminates before completion.
On a subsequent run the 1 2 - 2 5 card will cause a scan through
the Walsh array to the point where it previously quit and
continue from there.
One essential difference between the A and I modules is that
the I module will not recompute an approximation that has been
previously done; the A module always recomputes the approxima-
tion and replaces the entry on the RDAT file if it is more
accurate.
The K module causes a set of coefficients to be displayed
after rounding according to the strategy described in §5.
The first two parameters specify the approximation desired.
The third parameter, if not zero, causes rounded coefficients
to be punched out on cards.
The L module loops around the K module selecting the best
from each row of the Walsh table. The first parameter
indicates the minimum accuracy to be displayed and the
second parameter is the punch flag as described for the K
module. As well as the best approximation, the principal
diagonal of the Walsh table is displayed.
The X module may be used to clear out an entry in the Walsh
table that is in error in some way. The approximation
indicated by parameters 1 and 2 in the open data set is wiped
out without a trace. Parameter 3 must contain the key 9930
[a guard against punching an X card by accident). This module
may be used in conjunction with the I module to force re-
computation of a set of approximations.
- 17 -
The P module causes the error curves of desired approximations
to be plotted for display or diagnostic purposes. Parameters 1
and 2 specify the approximation whose error curve is to be
plotted. Parameter 3 specifies the amplitude (in mm.) the plot
is to have on the paper. Parameter 4 (between 0 and 150)
specifies the distance from the top of the page (in mm.).
Parameter 5 specifies the position of the labelling information
(in mm.) to be placed in the left margin opposite the error
curve; if zero parameter 4 is used. If parameter 3 is zero,
the last mentioned amplitude is used and the position is
further down the page by the distance 2 * amplitude + 2 mm.
Parameter 6 specifies the number of bits in the mantissa of
floating point numbers that is to be used by RASH for the cal-
culation of the error curve. This allows the effect of limited
precision of various machines to be observed graphically. For
CDC 6600 single precision this value is 48 bits.
The C module allows the user to delete or change the number
of a data set. Parameter 1 gives the data set number to be
deleted or changed. If parameter 2 is zero, that data set is
deleted. If it is not zero, the data set number is changed to
this value. Parameter 3 must have the value 9903. This is a
check against punching column 1 incorrectly as a C.
7. DESCRIPTION OF REMES2 SUBROUTINES
A number of routines are involved with the Remes algorithm
itself. The routine RIPPLE sets up and solves the non-linear
system of equations as described in §3. It linearizes the
system and calls SOLVEQ to solve the linear system. This
routine uses a Gaussian elmination with partial pivotting to
obtain the solution to a linear system of equations. The sub-
routine EXCHAN does a search for the extrema of the error curve
determined by RIPPLE. DELTA calculates the error curve for
EXCHAN using the current set of coefficients and the function
subroutine FN. N is a function subroutine supplied by the
- 18 -
user of REMES2 representing the function for which the approx-
imation is desired. If IOPT is 2 (specified on a B curd) then
a routine WN is to be supplied to determine the weighting.
Several routines perform miscellaneous tasks for REMES2. The
integer function IA does the indexing for the MS array and is
used in the calculation of the record number on RDAT. Routine
RASH calculates the rational function (specified as an array of
coefficients) for a given X-value. REFORM transforms a number
from its double precision format to a character string to be
printed or punched out. The assembly language routine WATCHME
with entry points GIMME and CHECK provides a constant check on
the time usage of various parts of REMES2.
The input/output routines provide communication between REMES2
and the file RDAT. Included are OPEN, OPEN1, PUT, GET, CHANGE,
CLOSE, WALSH, and PLOTIT. A discussion of these routines may
be found in §9.
8. THE DATA STRUCTURE
The program REMES2 uses an indexed sequential file (named RDAT)
to store all information collected concerning previously cal-
culated approximations. This scheme was instituted primarily
for the convenience of the starting value algorithm since it
uses information about previous approximations to construct a
guess that is going to be good enough to ensure convergence.
The basic idea of an indexed sequential file is that it may
be read and written either randomly or sequentially. In the
case of REMES2, random access is used when a new approximation
is to be inserted or an old one retrieved. Sequential access
is used when a data set is being opened (information about all
pertinent records is copied to the array MS; see below) and
when the Walsh tables for all the data sets are being printed
- 19 -
out. This is one possible implementation that can be used; it
is also possible to use random files or sequential files with
some clever programming.
The records are stored under an integer key that contains
basically three pieces of information. The data set or case
number specifies to which data set the record belongs and the
specific approximation within a data set is indicated by the
numerator and denominator degrees of the approximation. The
record number is then [data set number] x 1000 + IA(K,L) where
K is the numerator degree, L the denominator degree and IA(K,L)
the formula
IA(K,L) = (K+L)(K+L+l)/2 + L + 1
The value IA(K,L)=0 is used for a header record that contains
important information for the case represented by that partic-
ular data set. The following information is stored in the
header:
(a) The boundaries of approximation for the case.
(b) A word that can be used to verify that the correct FN has
been loaded (it also checks WN if the error criterion is 2).
(c) The error criterion flag that indicates the weight function
to be used. If 0, w(x) = 1 giving absolute error; if 1,
w(x) 5 f(x) giving relative error; and if 2.w(x) = WN(x), an
external to be satisfied by the user of REMES2.
(d) A message that is displayed whenever the data set is opened
and as a heading to the Walsh array display.
(e) A flag that indicates that the approximations in this data
set are in Chebyshev form rather than power form.
(f) A version flag that indicates the version of REMES2 under
which the data set was originally opened.
- 20 -
When a data set is logically opened to REMES2 (using the B
card), a search is made for the header record. If it is
found, the bounds are copied to A and B, the error criterion
to IOPT and the header message to MSGL and MSG. A check is
made on FN (and WN if IOPT=2) to ensure that the correct sub-
routine has been loaded. If no header record is found the
program assumes that they have been supplied by the user with
the B command and the header is set up using current values
of A, B, IOPT, MSGL and MSG. The reason for this method is
mainly to avoid errors due to the mispunching or omission of
data.
Thus when a data set is opened to REMES2 the above transaction
must go on; in addition, an array MS is set up to contain the
amplitudes of the minimax curves for all members of the data
set calculated so far. For those not yet calculated the dummy
value NOTAPT is inserted and for those that could not be cal-
culated for various reasons, the dummy value DEGEN (a large
number less than NOTAPT) is inserted. Thus the status of the
wiiole data set may be obtained by use of a few variables and
the array MS.
The most accurate approximation currently available in the
data set and its maximum error M, is copied to the array ZCOF
and used whenever the current value of M is greater than M-.
by a factor of 1.0 E-4. This requires that care must be taken
that M contains a reasonable quantity at all times, but gives a
large increase in time efficiency, since it means that high order
approximations are used as master routines for low order
approximations rather than the comparably inefficient FN
routine. The effect is particularly noticeable with the
P module, wnich requires 513 evaluations of DELTA for each
error curve. This feature is transparent to the user but
extremely important to anyone changing the code in any way.
- 21 -
9. THE I/O ROUTINES
To facilitate communication between REMES2 and the data set,
a number of subroutines were written.
The routine OPEN1 is called to open the file RDAT to the
system. It is only called once by either OPEN or WALSH and
only if the logical variable OPEND is false. Since OPEN1
sets OPEND to true, it will never be called again. The
logical variable NEW (which can be set using the N module)
directs OPEN1 to open R.DAT as a new file. The reason for
this complication is that the indexed-sequential file system
makes a distinction between new and old files.
The routine CLOSE is called before KEMES2 terminates to close
RDAT to the system.
The routine OPEN is called to open a particular data set
(data set number is indicated by MASK) to the program. This
involves defining A, B, IOPT, MSGL, MSG and MS as outlined
above. If OPEND is false, OPEN1 will be called.
The routines PUT and GET are used to transfer data to and
from RDAT and thus provide a convenient method of storing
and retrieving data.
The subroutine WALSH provides a resume of all information to
be found on the file. The header information for each data
set is displayed along with the estimated precision of each
approximation in table form. The precision is estimated using
the negative of the log to the base 10 of the maximum devia-
tion in the error curve. If OPEND is false OPEN1 will be
called. Thus to obtain a summary of an RDAT file without
doing any calculations no data cards need be inserted.
- 22 -
The subroutine CHANGE provides the features described under
the C module. It is used to change the number of a d.it;i set
by (1) verifying that the new number is free, (2) copying the
old number to the new number, and (3) deleting the contents uf
the old number. If the new number is 0, only step (3) is
done. This allows a partially executed change to be completely
recovered quite easily. The subroutine PLOTIT is invoked
whenever a P card is encountered and interfaces REMES2 with
the system plot routines.
Additional information on the details of implementation may
be obtained by referring to a current listing of REMES2.
10. SUGGESTED READING
The algorithm used is basically that presented in Ralston
(1967) with various modifications. The starting algorithm
is based on a suggestion by Werner (1968) and the rounding
strategy is suggested by Hart (1968). A guide to the litera-
ture along with description of approximation techniques is
presented in Cody (1970).
11. REFERENCES
[1] Cody, W.J., (1970). A Survey of Practical Rational and
Polynomial Approximation of Functions, SIAM Rev., 12,
pp. 400-423.
[2] Hart, J.F., et al., (1968). Computer Approximations,
John Wiley, New York.
[3J Ralston, A., (1967). Rational Chebyshev Approximation
in Mathematical Methods for Digital Computers, Vol. 2,
A. Ralston and H.S. Wilf, Eds., John Wiley, New York,
pp. 264-284.
- 23 -
[4] Werner, H. , (1968). Starting Procedures for the Iterative
Calculation of Rational Chebyshev Approximation, Proc. IFIP
Congress, Edinburgh, Vol. 1, pp. 106-110.
- 24 -
APPENDIX A
SAMPLE RUN OF REMES2
Page A-l gives a listing of the input deck for the sample problem,
ncmely the control cards, followed by the auxiliary function FN
and the weight function WN, followed by the data cards. Pages
A-2 to A-5 list the output deck produced by REMES2, giving the
coefficients of selected approximations. Pages A-6 to A-57 con-
tain the listing of an actual run, with the operating system day-
file on page A-58.
For each approximation computed by the A and I modules, the program
lists the coefficients, followed by the extrema, followed by M and
the precision -log,0M, followed by an estimate of the number of
decimal digits lost by cancellation. The coefficients are scaled
so that the lowest order coefficient in the denominator is unity.
The Walsh arrays give the precision and cancellation for each known
approximation. We define the cancellation for a polynomial as
max -log-|z
10
where the maximum is taken over the range of interest. For a
rational function the cancellation is taken as the greater of the
numerator and denominator cancellations.
RErtEiBNNNiCHUDaDOB.AT-MFTN.
LQaOCREMESZ)
r
9nnnai c ppcrfgTnM FnunTTriM
-THIS ROUTINE IS A- TEST EXftHRtE-EOft 1H£- -BEHES2C EXP«XI HILL BE APPROXIMATED WITH VARIOUS ERROR ;RITERI»C . IM--T4JE INTERVAL * 1 TO 3 -C
FN = OEXPCX)-RETURN -ENO
DOUBLE PRECISION FUNCTION MNIXI
C = = s » s THIS HN(X> SUBROUTINE GIVES RELATIVE ERROR AS DOES IOPT=1_C F-OR-OEMONSTR«TION RU»EOSE5 _
CDOUBLE PRECISION PEXPyXHN = DEXP(X)
END
APPROXIMATION OF THE EXPONENTIAL FUNCTION? " - ? *= 1 M
A-2
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FUNCTION FN CDC 6600 FTN V3.0-P320 OPT=1 1 9 / 0 1 / 7 3 1 1 . ' • ' • • 1 1 . PAGE
DOUBLE PRECISION FUNCTION FNCX)S ,C —== = = THIS ROUTINE IS A TEST EXAMPLE FOR ThE RENESZ PROGRAMC EXPLX1 WJLL BE APPROXIMATED WITH VARI3US ERROR J ! 8 I I £ S I A
S C IN THE INTERVAL - 1 TO 1C
DOUBLE PRECISION DEXPtXFN = OEXPIXIRETURN
IB EJJO _ _
FUNCTION FN CDC 66D0 FTN V3.0-P320 OPT=1 19/01/73 ll.itt.ll. PAGE
SYMBOLIC REFERENCE NAP
ENTRY POINTS2 FN
tfARiam,E5 SN TVPF BEL0C4I1Q1I -11 FN DOUBLE 0 X DOUBLE F.P.
EXTERNALS TYPE ARCSOEXP OOURLF 1 I THBtBY
•STaTT'STTR';PROGRAM LENGTH 13B
FUNCTION HN COC 6600 FTN V 3 . 0 - P 3 2 0 OPT = 1 1 9 / 0 1 / 7 3
DOUBLE PRECISION FUNCTION MN(X> ~
C====== THIS HN(X) SUBROUTINE GIVES RELATIVE ERROR AS DOES IQPT=1C FOR O£MONSTRATIQN PURPOSESC
DQJBLE PRECISION Q£XP,!SHN - DEXP(X>REf'IRNEND
FUNCTION MN SOC 6600 FTN Y3.0-P.3Z;] OPT=i 19/01/73 11 •'..11. PAGE
SYH3DLI; REFERENCE HAP
ENT*Y POINTS? UN
UABTARLFS <iN TyPF RELOCAI2DN11 MN DOUBLE 0 X DOUBLE P.P.
EXTERNALS TYPE ARCSDEXP OOUBLE 1 LIBRARY
SIAIISllCS-PROGRAM LENGTH 133 11
A - 1 0
//COMTROL// B T F f l
nF THF FKPnUFHTTai. FUNCTION
n«Tt-SFT- HFTNf. USFn TS 1
nnuNns nrcuBFn as A = - . 1 nnnnnnnnnnnnnnnnnnonnnnnnnn,-i»ni
B =
RELiTIVE ERROR CRITERION USED
/ /CONTROL// A t> 0
RATIONAL APPROXIMATION RfK.L l — K = »i L = 0
STARTING VALUES GENERATED USING K l - l j 1 )
-.innnnnminnooQnnoQnnooQoonnnnn+ni.30901699l>37<»9<il23<i013931S3<>70*00
. 7616D-01. <(.12
.1OOO0O0OOO0OO0OOO0O0OO0000G0O+O1
.99962 3625091.7531.6 73 098953 877D+0II .5 0 267 701650 1.11.0 16061.9 09619 l'fOfO 0
-.laOOOOOOD 000000 0630 00 Q00a.168 0181.6788 710168 9".552l.65"t 950 t . 7 5 3 0573 28 23 8196711.0 065 755062 D*ao .iooooooooooooooooooooaunoooco*oi
.59500-03 3 .23
. 999626519"t856 03 739218 30 2<l53aO +0 0• 17S'(8733'i 771625052105957115ia*oa
-.looonooonnoooooooooooDODOOoootoi
• 9979395l»26Z39i»62000358<.23a60D+0 0
• »56 82121.83 01037 0 32651.6601091.0 +0 0
.502 901.505 8413 86 Si. 1.86211.033700'^)
.11.90527 39 0595937011660B<f69 7BD*00
• 50I»OD-03 3 .30
.9996278963"il35997d97<t231»3999btW• 1761.662 31.934780 6751,7973 65371.0 tOO
.73868159213S'.611.0275Z 015902 00* 0 0
.997938726977751.88361939017530^0 0
.3996291659816850298590 927 9980-0 1
.laooooooooooDoaoooaooooojoouo.a:
.5 0289861.8 355515 9232991.6881(370 t
- . 1 0 OOP0000000000 OP0000000 0 0000*01.11.92368 3119921.9752Si.7l.99l.8l.3Dt00
-.8567908 03 0 929029220 76l>6287B5O»0 0.73864S1.5190975125169768 615630+0 0
.i>i.7l.lO8itl 70 5549260 961.71.95256D + 0 0• l a o o o o o o b o o a o o c o u o a o o o o o o o o o D t o i
.50300-03 3.30.87
/ / C O N T R O L / / 1 0 0 1 1 5
RATIONAL APPROXIMATION R(K ,L ) — K = 0 ,
STARTING VALUES GENERATEO USINr,~RC-17~H
L =
-.10000000000OOOOOOOOOOOOOOOOOD»01
.7616b*00 . 1 2
.1000000 0000000000000000000000+01
.61.8051.J73663B85399571.97735320+00
-.100C00000000000000000OOOOOOOD+01
. 76160*00 . 1 20 . 0 0
.100000000OOOOOOOCOO0030000OOD+01
RATIONAL APPROXIMATION RCK.L) -- K
STARTING VALUES GENERATED USING RI 0, 01
-.loooaoooagaooooooooooooaooooD+oi .539030285815795*8850*21727290-1* .tooooooooaooooooooooooonooooo*oi.76160-01 1.12
.1999959*922889075030*1163*890+01 -.375722730652*9001531338982*10+10
-.100000000000000OOOOOOOOQOOOOD+01 .187*3*77O5126*8695597725Z*11D-O9 .68*6**50**81*663556*8*0213770-01
.57620+01 - . 7 6
-.*3309i97377566B3265655906505D+01 - . 1689262 8*8*O96311O671921135*D+11
-.685080016*7166709312*19*97910-01 .785>>ai03312Z979710199920ia36D'10 .1000000 00 00 000000 0000000000 004-01
.1225D+02 -1.09
.i30*215U3627*9530270777Z616D-0Z -.lZ72*a*Z9736588*0*S**909S76ZD+ll
-.10000000000000000000000000000+01
-2.1.11
.7B591Q0 0<i762it317B67Z05692076D-10 . 6B»6<t't50655 97 25 72166210120*70-01
O-l- . 361990*10 5310 P32"i5865B5D 78660-02 -.lZ7"l71iil7811if93597if 5632701110+11
-.68508001813516103651190551800-01 .78*98895919890* 17917*1.27321.90-10 .10000 00 00 00 0000000000000000 00 »-01
.I.6B7D+O1
.1<.*6096**7091267*266*IJ751865D-05 -.1Z73902 3*631107978*9868 217 29DH1
-.1000000000000ODOOOOOOOOOOOOODtOl .784989525610067311*9793312950-10 .68*6**50655 961010155 0618a*6liD-01
.96580+03 -2.98
-.*005*9069679*57081315*923S8*D-05 -.1Z7390*900*6B177380773303701D+11
-.68508001813673903156*91662900-01 .78*988*96620381760597 57177390-10 .10000000000000000000000000000+01
.*783D+01 -.68
.1601151027183187081.0390757070-0 8 -.127390*0170712*8*80*723*37030+11
-.100000000000000000000O0OU0O0D+01 .78*988*9725600593*329362777*0-10
.17600+03 -2.25
.68*6**506551917 2*8*395*153600-01
-.**3*96193623351082J23952*6l6D-08 -.1Z7390*01988550339111935SB69D+11
-.68508001813932038785985606**0-01 .78*988*95089502962*0*319*0230-10 .10000000000 00 0000 0000000000 00+01
.5056D+01 - .70
.177309132053829175*5855511300-11 -.127390*019951501858*618 3")8*ODH1
- .1000000000000000000000000OODD+01 .78*988*9508997*6980*20S961180-10 . 68*6**51*6936107*5321Z55*94»6D-01
.52010+01 - .72
A-14
'3 !
a
i
=1
i 5
I !
! £
SI SSi 3-rt|Wlas; ^I
a
I 3
a s
ISiJ-l
3
3CXI
J 3
. r
.50560*01 -.70
n-ii -.1773qn1.ru
^J- •"•«.7%'t396795070Z535j.3ZZa 86695040+00
1760D^iz"
.100 0000 00 000000 DODO 00 00 000 D0O+01
Z.7S
.52nlD+ni - . 7 2
•*_fiA5nAnni ft8*5fl fcfiufi'*?^ •wstdQi non-»ni
.63800+01 - . 8 0
**ERROR** REHE5 ALGORITHM NOT CONVERGING
RATIONAL APPROXIMATION RCK,L> — X = 1 .
STARTING VALUES GENERATEO USING R( 0 , 0)
-.10000000000000000000000000000+01.innonnnnnnnnnnnnnnnnnnnnnnnnn*ni
.7616D-D2 2.12
qqq7Rl.n<.fc7fc(lU17T«.7nSRIl6l.fcn0n+OD
-.iffnnflRflBDOanflfinnnnfinnnnnnnonn+ni.lnooooooooooooooaaooooaooaooo+oi
.20820-01 1.68
.9997631.352351. 781(86579967 93 52D+00
-.iDOooaoaoooooooooooooooooaooo+01.lnonnnnnnnnonnooonnnonoDOOOon+ni
.9fiiiin—tii i .[.A
• * *
PITTRMM. APPRnXTHATTOM R«KrLi — If = 1 .
-.itinoonflaonooDDODflDaoonnflnoaan+ni.718393677611.9160152726835601.0+00
.31.1.00-02 2.1.6
.99830251182833926790950757100+00
-.I273qn^niaq5<.«f,7i«nir.36i.i7n+ii
7Ri»qK»i.qf>nRR5fiinqim5q?qiQi.in m
L = 1
-.5000000000000071051. Z73576D10D+00
.47nikqll&sqon36fcR6ni91RR?17in+nn
- . 5 1 a 1 r>7t> if,?3nf>f.n.ii9.i 7657787 53n+no
.'»70160Z75c579a6aZOZZ<.<.6585'.80+00
-.51SI>65ZI|Z690 783058811928951ZO+00
L = ?
-.71R3q3(19176117iqqq661SSl.S»7I.O+nO.looooooooooooooooaaffooooooooD+oi
.3253996881.1972537716576091.660+00
.innnnnfinn/lnnnnnnnnnnnnnnn.innn^ni
. 1,999999999999961* 1.72863211995D+00
-.u7n?5nfifiq7R77^^ai.3>;?RiR7ifif,i>+nn
.5iain5fc?5fla?f,R977m.77qii!n.Rnn+nn
-.1.70 262117151735l.570Q8m672bOD+00
.5181.6521950 51.93 385160890567 70+00
-.13557078953 72 5861. «.2l.snS85nBf.n-fl7
-.67ZZ<><l90<l30625676d<tZ98l.631ia(iO+00
• 99833Z959*69i.3979385<i73683Z0O+00.16006298*79821081531.291125 DiD+OO
.32S1337663'.256908(1509230955 bG= CO -.&7ZZ333Z9970391Bi76795TSS&a3D+00
- »10flHOfliPOiO 0 OJO 0 q PO nonfl Q D 5 a CD *a 1 - .67873016sa76in8320S'H]67i .a i3SO»oo• r<>30023'>2266<t't9593S9<»7<»23'>760*00 .100 0000 000 0000 DO BOO 000000 JO 00*01
.17280-02 2 .76
RATIONAL APPROXIMATION R<KiL> — K = 2 , L = 2
STARTING VALUES GENERATED USING Rl 1 , 1)
-•10000000000000000000000000000+01 - . 8 1 6 51678622891.50 79528 7B<»Sli50D*0Q -.321234781.10 87^281.3500 29<ia390DtOa.JglZI»7Sg<i3l63g62766Zl l 119010*00 .816516776B367B51'tOBSB'il0l591D*0ll . 100 OOOOnOOOOOOOOOOO 00 000000 IIP 4-01
• 56J7D-03 J.25
- .9qq99«iqqS?3a666l.5176167676710*DO .<.q9785565175»O9377037l.6O'('i0OD*0O . 81Sq2SH.3636771.115469731ZZ98D-D1-.V9978556702352875755310 751.580 tOO .81592511.6562607869802 65196940-01
-.100000000000000000000 00000000*01 -.812OO't3<tS3ai331O2137!><»17272ZD+OO -.31Z07973Z166013<.77193'.570132D4a0. .T«?07q7aq5597626713iB7q?f,9B70i-nn .m?nnh.^';i7gS7qfcyftq3RiHSgsi.7in*J0 .ninnnnnnnnnnnnnnnnnnnnnnnnnnnnn
• S6S3.U-04 "1.86 _ - . _ _ .
O\
o
A - 1 7
k I !
I I
i !
//CONTROL// I 2 - Z Y 1 11
APPROXIMATION R1K.I.I — K =. t i _L_? fl_ .. _ _ . . . . _ . . ._ . ._ .
iilB37i917851lf3&56ir73M(565390t01 .9Q15212Zl.5832l,32a9922l.3126a60+0 0 _. . _
-.lonooooftODOOOonoooooooonnnooo+ni - . 3 1 1 65*.7?n?qgq7fl5t»?ini7iBfl(»i.qn+nn • innonnononooonaoDnoonnritinnnnn.-ni
. 87
RATIONAL APPROXIMATION RtK.L) — K = 2 . L = II
.111387898877^9825751.9221 TOfclfl Hit _jJt6a35_1261it537 0650 719131027 a 1D*Q 0
-.10DODOOOOonQnnoQOOQDHnODODOQD»ol.10000000000000000DO 0000000ODDtO1
• 397'tO-Ol 1.1.0.83_
RATIONAL APPROXIHATION R(K,L) — K = 0 , L =~ 2
.97211.2i.66<. 1.511.71786129282317D + 00 -.1081.58160698330 01565833231590+01 >i.570079li.2i.52a9'.720376773659Ot0D
-.100000000000000000000000 0 0000+01 - . 2 3 2 2283158336370371.07820031.60+0 0 .651|8837393906l.9l.2a53l.7S0<tl61D+00.100000000 0000000000 00 00000000+01 . ._ . .
• 3975O-01 1.1.0. 83
RATIONAL APPROXIMATION RtK.L) — K - 3 . L = 0
. 996589611618279516585 7630 6700+0 0 . 101 QgQ36ll61.7;?63 03731.63579381O+0 1 . 53HBI.q635121.3 91565363 0?21Bli8n+ 0 0
.1585170231392392630 7557339290+00
-.1000000000000 0000000000000000+01 -.78909572722828393553628981880+0 0 :rTl95177699231i.9~5 0 0 8 81*943713 2 0*0 +00• 5851091i.030357726031'.60761'.5D+00 .100 0000 00 0 00000 0000 0000 OOOOOD+01
. 87
RATIONAL APPROXIMATION R(K.L» — K = 2 . L - 1
»»FRBOR»» MAXIMA OF TgROR CUtitfE OESEW6RATE
RESTART ATTEMPTED (SIGNI RESET)
RATIONAL APPROXIMATION R(K,L> — K = 2 i L = 1
" E R R O R " MAXIMS OF ERROR CURVE DEGENERATE
RATIONAL APPROXIMATION K « . L ) - - K = I , L = 3_
. 1003l.77l.867021511121.362191.880+01 - . 101 »344061.99325591587578293.10 +ill . 51.073701593499373732B7Q51.3550 + 0 0
-.1590722<t52025S9566Z6Z12790'»9D+00
-.13000000000001900000000000000+01• 7amn37a7q33fc31q7.Tg151n1iRq73tn.nn
.585108928769'.13i.4<.31367760bi.D*00• i nnnnnnnnnnnnnnnnnnniinnfinnniim.nl
.l95179B3957D8'»19S'«5332152a00Q*00
RtTTONAL gfK.L» — * -
.753 g35O67fi6ftfl 3a7qBgfc3fcfrm7Hnn4.fi n.I.03S659536239209077796721956D-01 -.2*738901176<t719805525756Z3l5D+DO
-.lQOOOOOOODOOn00000000DDOOODDD+Oi -,B37DZi»99063i«33i*07362>.3BOB5i>70<.0 0• innnnnnnnnri nnnnnnnnnn1mnnnnnn4.nl
. 65
BtTIOMAL APPROXIMATION RtK.LI ~ K » 1 , L - 3_
7iF.nqn.mn. 25238501(830391.0 "tl69l6*7910<i9Dt|i0
-.100000000000000000000OOOOOOODtOl -.78D'»855bl»25lf6B7 191680 67535710*0 0 - .233 2521.71161778J0067 0 ".0621.060*00.innnnnnnnnnnnnonnnnnnnnfifinoGn+ni
.12910-03 3.a9. 65
UPPHnXTHATTOH RIK.L) — K = _= i_
-.176551928 315517766"»20a253116D»00 .39977790^1832ZO31179335697260-01
-.10000000000000000000000000000+01 -.738701988 2239250 63257935611.50*0 0 -.11.923 62510881.21.758105760 31.79D+30-1nnnnnnnfinnnnnnnnnnnnnnnnnnnn4.nl
.gn3nn-na un_.87
RATIOMAL APPROXIHATION R1K.LI — < = »• L
• 671970566298311. "409855391111.7D-01 .8110376011551336139363062356D-02 -.1983515188D550363965970210020*00
-.1000000000000 00 0000 OOOOOOODOD+01 -.8a611229^.^7t•3l.22ftll7522 8661.60*00
.looooooooooooaoooooooooooooootoi
-.5619681(67 6635S2<t758907B95820Qt005a«66na?B7fl';qn«.nn
.85863-05 5.07,63_
RATIONAL APPROXIMATION R(KtL) — K = 3 , L - Z
.10000B<iZ71<>13B9167«3%0<l65li 660*01 .6006131.71797161.2271500751.9270*00— 39q3a1.fiq76ll6q1.qftq1.q2n7n1.SlBSn.nn
.li>99Z92713fiB<.20751i35Z10519870»ao
--tniniiiniinnnnniinniinnimnnnnnnnnn<.n<-.2865951. 907S93S9701".'.66092983D-01_ .iooooooooooooooaooaooQooooooD*oi
• «.8060263610899256'.919310l.526D*00 .860088531961.337 9398 D78Bi.1071Ot.00
• ••335D-Q5 9.36.52
RATIONAL &PPSOXIHATION R(K.L> — K = ?. L a 3
.999995728552250 909960 4905072D+B0-.600610833 006253220 767 690 1.473D +0 0
-.1000000000000CO 00000000OOOOODtO1,2ft6B'.9027'»2161H.Onioa01.0n5010-01
.149928581297719955139665270 00+aii'iiisfi 3i 3 is r 2 &zs svzitaso -ai
- .16347274 W 9962 70 37"»679 7^753630 -01-.86029317929363673606179562550+0 - . i . 8 0849651.06 3613 6<»0561258'. 1*7 20 v
.iooooocooQaaaaooQooooBooooooo+01.it3350-05 5.36
.52
RATIONAL APPROXIMATION RtK.L) — K = 1 ,
.10 000 0747780118344546 786854'7D+0"l
-.iooooonnooooooooooooooooooooo»ot-01
.100000000DODO0000000000000DOO+01.5619681.69893765 2281M 71937990+00
. 8 5 8 6 0 - 9 5 5 .07.69
.386112285121322351693 6<tO192TQtO0
RATIONAL APPROXIMATION RCKjJ.) --.K * _ J j _ L _ 5 . .5.
.999972«.3067<t91251'.'.663533a3it0+011- .167269 61*7690179 56<. 1)097696392 D+00 . 1,361.505557713652886180 3l»3SaaD-Q
- . l a 00000 030000 0000 00 00 00 00 0003*01 -.S2297D9 55327ltl316883a67612<»0D+0 0• 11.10265 827793130127ai.»6Z37250+00 ,59618633003693570117887116620+00.10000000000000000000000000000+01
.a961289060'tO7't2171.390212 971.10+011
If.38.87
RATIONAL APPROXIMATION R(K,L) — K = 5, L = 1
.999999719<»853<t716«565059182D+00
.83652233951 .75179879H.85H.5280-81.833 9122 775031*73311.85380 303570+0 0.13q78795061g6682857972681665D-01
.3339198690918 70652156231.05630 + 00
.135557138 60 6<>26.253611610Blq.qi3-l)2-.16608I*51it7l>79<.75051269<.925380+00
-.lQOOOOOaCOOOOOOOOOOOOOQOOOOOQ+01_ - . 2 9 9 8 1 2 6 1 1 3 6 5 6 5 7 8 0 2165198fcei^P+an
.88393981158189671599713285670+00
- .9153290913956558010323695731+0+00 -.671431590 88 9386 95490567260500+00
. 10000000000000000000000000000+01
.51090-06 6.29.72
RATIONAL APPROXIMATION R«K,L) ~ K = <., L = 2
.1000000059171.75 029119329l73&a+01
.33301.606; 98758.667i;<>'.1958062i>i»5lia997<i95'.6100+0 0
-.10000000000000000000000003003+01-.2524899720 01.91.506m82 91.269">2D+3 0
- .90902178 0 0206090 288 38 997768 50+3 0.183 252998797980 20731.6910B091.0+0 0
.Z0D220313521.2 81362393 39707600+00T 3,1215 7211291566 9568Jll3Jt|JiliiL3at(UI
-.61.9511.656 07 01296950 50 62655 740+00.5 9880 38355134462021.50 52Z3123O + 00
.S93505ZOVJ5E>1259870B92916i&50*00 .10000000000000000000000000000*01
.20610-06 6.69
RATIONAL APPROXIMATION RfKtL> — K = 3 i L = 3
.99999999999998812386174173840+00; fc5^Tr7gS? 7 7 7 s n ?
-.8229453047Z217793493791772510-02
.49999946850 775975976832289790*0 0--..!1q.9.9q94fiP'an7l)6fl'iM1257r'546a7n*00
.997835473950927348994aZ118Z9D-ai
- .10000000000OOOOOOOOOOOOOOOOOD*01- .27351767? IiRllfiSS?n6Uq5U?t)?7Sl.n*nil
.9016791627676284252227534119D+00
.1550D-06 6 . 8 1
RATIONAL APPROXIMATION R(K,L> — K = 2 ,
.999999940 825 2 D9"»623 8955873*20 tDO
. 2728 C<>Z27710B 1)74577567910 9<>5D-02
-.HJOGOOOOOOODOOOOaOOOOOOOOOOOD+Ol
.909021769009891>626762'i739698O4'00
.2061D-06 6 .69.58
RATIONAL APPROXIMATION R(K,L! — K = 1)
-.90167916275159902106098585190+0 0.??3S17672371.103IH1(iS91lt77797?n + nn. toooooooooooooonon00000000000+01
L = <»
.33285719860377751717325970*OD*0O
-.893505216«»7»5D't9585it71727657O«-00. ?62 hgqq 7ni.7 .ipn<.7iq.77(iit7qQTQn+nn.iooOODOoooooooooooooaooaoooootoi
L = 5
-.62522221Z<i7321l(lt359J8396<t53*D*00-f,?<;????i?saf,ai5i.ni?x7ni<;?Qaf,un»nii
.330780652Zl<t2SZ08Sa539532<>lBO-01
-.598S038'tl>'>S8ii9300ia331266a3aO<'00-<iuqi5iufikSnR<»77fisi.3is?.-»qiianRf,n*nn
.100000028051593715826862118901-01 . 166084564("17821757"t8S1960933D«-0 0 -.83391250805107574532709614320+00
-.135557162768003051308581108100-02
-.1090000000000000000OOOODDODOD+Ol -.88393977167531139842710 38445DJ-0 0 -.569656397738680 015110 75048530+00
.91532893183248987084211571890+00 .1000000 000000000000 00 00 000000*01
.51090-06 £.29.72
RATIONAL APPROXIMATION R(K,L) — K = 5, L = Z
.99999999147234857431305714030*00 .714614804147225536046233894 00+00.594484781917001579037669149qn-0Z
.23830277654070029510049Z49090*0B
-.285385216956553B39S553892696D+00 .23687705772454796147964419590-01
-.100000000000000OOODOO0000OOOD+0130
.68319699493139126730103762250+00
.91850-08 a.04
-.9307540 90731288228158 22548450*0 0-.4760196253497706742631.9347770-01
= 91686990018199". 55829226873320*0 0
-.7306B530757ZlZ347731B9358516D*00.341911929460749732860 99437ZSD<-00.loaoooooaooooooooooaooooaoooot'ai
RATIONAL APPROXIHATION R(K,L) K = 4> L = 3
.10000000054B874S8789Z4872238D+01
.18997899033168357000664640620-01.5715609850148409026Z39102B110*Oa.117S7010593346736651Z9590407D-02
.14283303562939594395340639940*00-.42843901254550615040245967100*00
.7127222*735673772*29890811200-01 -.*721**671ID 595*7*5**806101260-02
-.loooooooaooooooooooooooooaooo+oi-.397*2161q955117qq2*ql731233*D+00
.7005179*6903758 26122917*0 7800+00
-.92666053183976512109*97525230+0 0-.15915*56**2*B6Bian6n7i7:>:>8fian-m
.922080552752555597181(53 578890*0 0
.71627327867IB220 9196639353930 + 00• 37G332B63*6*flSq3*IUflfl672fl68?n + []n•IOOOOOOOOOOOOOOOOOOOOOOOOODOO+OI
.55330-08 8.26.50
RATIONAL APPROXIMATION R<K,L> — K = 3, L = It
.99999999*51125615798266798Z10+00• *721**66qSl*33*96l5*?'H)53*38D-n2
-.189978988987882*180092*92093D-01
-.100000300 00000000000000000000+01-.37033358Ba3063H721*33H887l*:'[l+nn
.428*3901039552631585690935790+0 0 .7127222**251567*79727**157400-01
.11757010*96216*3511893308*180-02
-.9220816398*710203*0 82*6*81520+0 0 -.700 519736826*1*502*6171612530+0 0
.7162719956****590*85677605580+00
"755330-06 sTzY
.92665992950901*52*50316*96870+0 0 •loaooooooooooooooDooaaooooooo+01
RATIONAL APPROXIMATION R(K>L) — K =
.1000000008527651*223787699010+01_ -.71*61*810 23*63905739*250 86390+0 0
.59**8*7869*100287*06*5*720090-0 2
2 , L = 5
~ rz853852i9396792T21*0996232*60 + 0 0
^•739013*135987023998758*6296770-0 3
.2368770 597573 71*7911 6183 3**80-01
.-t7fifil6fi7113i:,a361735,iO99S*795a-Dl
-.lOOOOOOOOOODOOOOOOOOOOOOOOOOD+01-.3*191192931158*8768603516707D+0 0
.7306853080822*197228155885020+00
-.9168699 00 0526 51*5*0326593*170+0 0.*76019627*76zn***3l009702*63D-01_.930 75*0 9128**85*13216011370 00+0 0
.68319699*759*0 3 0176 5221017150+0 0ii2iA3ZZ.Z15.Jl feli2£9 9 7 63 3aS_3 79 60 • 0 0.1000000 00 000000000011030000000+01
.91850-08 8.0*.62
RATIONAL APPROXIMATION RIK.LI — K = 1 , L = 6
.999999977982*0761930*20033210+00 .1*2531*670679*93*111128750330+0 0 -.857*68*005*06793026*3756J72ID+00
-.2393551230723*271023980918**D-02 .19*076*0919366372678*8*259060-03
-.100000000 0000 00 000000001100000+01-.31259578905669762950933697130+00
.7**0*788216587*53582279/30010+00
-.911111069*1966589297765967210+0 0.79036259*0*09353*3317*5759*50-01.93**7*57292698727*2096*080100+00
-.66*6191**1919*3 33*2*0 091*2250+00•**775»12B30520620110772535920+00.100 000000 00000000000009000000+01
.27350-07 7.56.7*
RATIONAL APPROXIMATION R(K ,L ) — K = 6 , L = 2
.9999999998338*728518777272*20+00
.59572692779581988 010030 20 5**0-01
.*a8125*0776655259376535*63320-0*
.75022559137526225275782861**0+0 0
.893*177369836*573822*00*5320O-Q2-.2*977**05*09370*1615876579280+00
.268018782 9198*3*0 2017*7775260+0 0l 91*1192189870963732*21*5 *36O-03
.17793183919*0230622501190735D-01
-.loooooooaooooooooooooooaooaoa+01-.5368**9585*961*90 88053823 7770+00
-.9*53503526162***1332272289190+0 0-.2218160606076821067*990 657180+0 0
.7861558360131670853 8612630890+0 0
.12*91326809530572101189687720+0 0.*619089711659891*72B737032260+00.10000000000000000000000000000+01
.7**8788212*16*78100 5519869330+00 .3336617600736*530783252123*70+0 0
.38230-09
RATIONAL APPROXIMATION ROC.LI - - K
.297*99B8520S76s79509*035BB020-01ni)
•2967*18Z876Z*5339*5*1Z93*6590-02.53* 71)8327*73
.1*69929565710699926897*1633*0-03
-•mDigommnnonoDtinnminnnniinniinn^ni-.519*7526*27653*7975277031*250*00 -.l9BZS9'>5525<i21389Za<>i>29B2&81D+0(l
. 1 0 0 0 0 0 0 00 BO DO DO O O D O O O D D O O O O O D * < J 1
.19200-09 9.72.51.
RATIONAL APPROXIMATION R(K,L> — K =
.100000000 OODOOOOOOOOlaO63558D«'Ol
.118725f,U35ftq05i.3?qilq8q5flDliqO-01. .1070761(1*35839911.2265213258300*00
-.10000000000000000000000000000*01
.50 0981.71I5950 372 75971.90751.0 61.0 tOO
.inaoannQODnnDonnnQonoonoDOOODtoi
.153BQ-04 9.S1. 4 3
BJTrnHAL APPRn»THATTON RIK.LI - - K =
.2958Z1.96951.91S1.699I.293333095D-02
— innnnQQflQppnQfiofl'nfinnonnnnnano+ni-.<l8 2031.9512998229891. <»0 7128 827D+00
.100000000000000000000000O0O0D+01
.19200-09 9.72
RATIONAL APPROXIMATION R<KiL) — K =
.1000d000001661527131li77220830+01-.7>;o??'i5qi<.q7iitni?fin6Q2i.?ni.2aD*nn
.893'tl773711'.60261»60'i5101ll929D-02
-.10GOOOOOO00000000000000000000*01
<*>
3.
Z,
L = i,
.<i9999999931101<t7613196<>722ltaD+00
.Bflqq3ailli.322S777lai.BSql.i35?n-ll.T- . 118 7Z561»3 58 7527 836623 00 79&66D-01
-.939903166*5*558 0913 M8B83Z90O4-0 0- . I 7 * i n ? i 34q3i n3Q75.tqi^i ?«;?i 1 .m«-nn
. 7667930 Z0 68353512t,1213633!.23D*0 0
i = <;
. 37i.R«;Hi.7fiun?7ifii af,n^a?<;_iif,f.n*nn-.6251*1525Z6017559*118*l|7130'»D«00
-.q17n17q777R7717nn3A.f71.5q1i> 7 ?n*nn-.11.971189393221.1.1.1.90 071173599D+00
L r. 6
.2*977<>'tO5Ci529ii52532B8999S819D+aO.2ea01B7B296261>292n563l>A275n9a+aa
-.891l.3192201730335300629'.l.lJ9D-03
-.933661758999B51.3780 0310*16850*00- . 1 ?*qi3267i»&n 117831 iiAin^nxi nqn+nn
.107078**358*75633051511*1*380*00-.<.qqqqqqqq.inqfcHaqus?3ii •m»qfiin*nn
.589938111*19668*0869215165150-03
-.7667930206*1613976678223388*0*00
.939903166*9692955052758813890*00
.5^l>7flHt771.57 7117Eq7nn|^n?l.«7n.0f
.17861235563*88578316B39120*3D*0D
«.7C!Ab1 7&1Q11 .15AQI.1.AA1 H.Y7A5qRAn*nn.198259*55281*0052660560697320*00
.17793183922700320539318679590-01-.5q57?6927fl878l.7?77517qi.fl?86qD-rjl
.*8B125*07a3982ZllBZ*7B370298D-0*
-.7**878819*28779311560*39B9960*00
. 5368<i l,9617a<.60l.7959777ia7368O+00•iooooBnoooooBnnnooooooooooooD»oi
.7861558'>179D5't27&l«6D81219<>804'DO .9I>5350355<»B05Zai998167351877O«'Da
•38730-nq.65
appgn»TMtTTnw H I K . L I — *
.3968776678612050707319*778880-01
.16337232286807780*B211616777D-0*.*957ia9*5*5*D80*27S3056169520-OZ
-.333Z1181ZZ6708071S5*775353110*00.395558Z55B*15B0*39859*630556O-03.*160Z35796737126792*316765330-01
A-24
l a S ) 9
35
ani I
s
CT'N.
1 IM S
SIS inT4 Kj iOin in toHSIN
K tfJ ^
H aCO -4tD «CCO tfni if(O l\• M(T a
CO (TO CT
OT CO
tnNindcrl tn Ki
B O M«fn IT-l 0J J
o c\o c+ I
-.1000000000000 0000001.000000000+01 - . 9460197 73014487544110 3 63Z279D+0 0 -.7909281370183 SO S3S2142Q50737D +00
.35402294378315672596059083270+00 .61995382441421125394487219990+0 0.1 nQ nnnnffrirtnnnnnnnnniinnnnnnnnn+nT
.42583178981652895060694079660+00
RATIOH«L RtK.LI — K » 7. L =
,69386569459 6453 77603409211690-03
.85097967072555681392808269090+00
.6358714493873586984819217377D+00-mnnnnnnn,in nnnnnnnnnnnnnnnnnn4.nl
.19774848490748620785727468910-03- fiftfi?ii7u?*if»r.?fi9<tp^9-^^flfl^-'5i **^n*0 **
,84649492004335542810337794650+00
.645971740B204016499550292440D+00-innnnnnnnnnnnnnnnnnnnnnn.innnn4.n-i
.48622876113009046497239491720-01 .69444757711S6237658132665236D-02
.33291451827746121850673732600-01 -.13842539791427192303910515410-02
'.10000000000000000000000000000+01 -.96213129325282392017343167270+0 0
• 109655676805273B437098511553D+00 .3878622821470484609703015055D+00
_BA1IQNAL B I K . L I — K = 6j__L_=
-.5S<iltZ6651ZI>70E1067850'>56l7660-OZ .19742124369 0618 3211.3011703190-03
-.10000000000000000000000000000+01 - . 960 9283 884485927839 06<tZ 391910 +00
.1Z619I.2839I. 429623872571231200+00 .40Z18497466418Z3101B930738490'»00
.11640-17. 5 2
OtTIOMAL tpPROXImTIOM RIX.LI — = 5 . L = S
• innnnncooonnnoaoonnnnnaniigi27D+ni • 4999999999994847627978 n6n?qSD*llO.13875044958952720363300660540-01
-.49999^9999991.4961295917717160+nH.98929812138681714463347745010-03 .3Z83969886909256995B644Z7863D-04
.13«7504495Bq4a956S67R96q79?9n-ni.98929912138639982306503048590-03 -.3ZS39698B6907183318748640ai7D-a4
-.100000000000005 00000000000000+01-.6554614775D2382gQ5D97965296aO+D0
-.95955623707066533764936133350+00-.4160n725166610321B9B3a6SZ573D*00
-.81.1577591.2574885831225616664D+00- . 142 5595an 89 9on88B9719351.9133D+nn
.14255958090112951466840780670+00 .41600725166764188269625214730+0 0 .65546147750392831950959343100*00• HinQfjnnnnnnnnoDOnnnnnnnnnnnnn4.n1
.43
tPPROXIMATIOW BtK.H — K - 4. L * 6
.39994392167 ?8 4516488M R723JI404.D1I.5S4426651Zi,719&0fl4637!H65183O-a2 .19742124369069527507456054170-03
-.277711BlB3q48174aq4149q967ll0n-Dl-.60005607832840283406141324070+00
q-,1977ii84849D74005387895042i640-03 .6553386589059629696117107926D-DS
-.ioooooooooonooaooDooooooooooD+01-.64597174081976573971039718580+00
-.958303082267539280187167411704-00-.40218497466317438482897766840+00
-.B3680530981178928614131087550+00-.12619428394324691870192553370+00
A-26
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CSJ I f
C\J CP J
in w ain tn uvfl in o
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//CONTROL// K k 0 1
lPPBn»TM*TTnM nF THF FKPnHFMTTftL FIINKTTOH
FT BFIMG ll-igD TS t
— innnnnnnnnnnnrinnnnonnnnnnnnnrn-ni
• inanonnnnnnnQnniiflnaaaannBonnntni
BELJTTUF EBROR CHITFglOH USEI1
PFPQftT Fflff APPPnXTMaTTnW 1H h- ftl i-Lnr.iniMii =
pgnnpn MIIMHFP nu pnaT TS i m i
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0)Q?0)
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7Qfl873S656Z3
//CONTROL// L 3 1
APERDXIHAIiON QF IME EXPONENTIAL FUNCTION _
JMI/tSEI BEIHGUS£B IS- _L_ _ ._.
BOUNDS DECLARED AS A - -.1OO0OO0000OOODOnOOnOnaOO00000*01
8 = .iflaOOOOOOOflBOBBaaOOBBOOOOaOOD+Ol
RJEJ.*TIVE £RROR CRITERION USED__
REPORT FOR APPROXIMATION Rt 2. 21
RECORD MUMBER ON R0»T IS 1013
PRECISION l-LOSia(Ml)
ESTIW»TEO LOSS IN PRECISION DUE TQ CONDITIONING IS ,1. DIGITS.
21 .1ZZ56 02619POtP02
aooQ01at z
II .61251 8500II .lOOpo OOOD _Zl .12256 02623II - . 6 1 2 5 3 BSOZI I .10000 0000
00
REPORT FOR APPROXIMATION R( Z, 3) PRECISION I-LOCIOIMII = 5,36
RECORD NUMBER ON ROAT IS 1019
ESTIMATED LOSS IN PRECISION OJE TO CONDITIONING IS .5 DIGITS
POOPllPQZon a
Zl -.611/2 015<i6 821 -.2*431 1T13» q
Q91an?
II -.30213 8690821 -.81ITZ 27&76Zl .367*0 73210 3II -.91711. T?STO
0» 3 II .10000 00000
•0-REPORT FOR APPROXIMATION R( 3, 3) PRECISION (-LOGIOIHII - 6.61
RECORD NUMBER ON ROAT IS 10Z5
ESTIMATED LOSS IN PRECISION DUE TO CONDITIONING IS .". OI&ITS
POO C 31 -.12151 «7585 ".61PHI i ?) -.unrgT 31371 RHPBZPI13
21 -.12125 173Gb 86i l - .mnnn minim n
Q00amao20H3
3) - .12151 47585 *>6171 .filt75T 31371 ft721 -.12125 17366 86i l .înnnn nnaon a
gfPflgT FOR APPROXIMATION Ht i. r 31 (-i osincMii = 8.26
FSTTWlTFn LOSS TW PRFCTSTnH DUF TO KnnnTTIONIH& TS nTr.IT?;
P O l <P0 2 ipa 3 iPUk iQUO <
Q02 <ni i
1 3)1 211 1>1 011 3)
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-.12105-.30251-.VÛÏÎ37
-.21179
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632<i<>
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it20760 00 DO
6852611.31a%260717769DO ID
REPaBT FOB ftPPROXIMOTION g ( «.. <il PRECISION t-L.OS10<H>) - 9.81
RECORD NUMBER ON RDAT tS 1D»1
FSTTNâTFn l(l« TM PPgHTSTOH nUF TO ROMDITÏONING IS •fc DIGITS
PODP O lpnjP03POh
oat002an 3
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31712)11
[ <>>1 31
311 79
. 16950>8<>?5>l
. 2 0 1 2 5
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.18150-.?01?5
6S3017Q76810147onoDO93062
79268iniU7
5li77l>05980ARq.1194320 2151.77".
J1338A67631»4173
7
7
II .10000 00500
REPORT FOR «PPROXIHIITIONI RC 5, it) PRECISION (-LOSiO(li)) 11.37
RECORD NUMBER ON «OftT IS 1050
E TIHATEO LOSS IN PRECISION DUE TO CONOITIONINC IS .5 DIGITS
POOPS1P02P0 3
3).16912.42271.
7926335692
52348 41H5J91 W73283 9S1767
P04P05aooaoi0.02
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-2J.
.50176
.19949
.301.40- . U 5 2 7,.25349
81272802555M.Z1.75161151B715ZA1
626137351553 8505577 362D373 9••B075
Q04 < II .10000 00000 0000
REPORT FOR APPROXIMATION Rl 5> 5) PRECISION I-LOS10CH>> - 13 .01
RECORD NUMBER CM RD/frTs 1061 ~ ~
ESTIMATED LOSS IN PRECISION DUE TO~ CONDITTONTNG IS . i " V
POO_PO1P02P03PD<vPD5
aoo-Ml902•D3
as?
[ 5)
3)I 2)I 11
5)31
312>1)
-.30450-.15225-.33825-.42250-.30125-.DllliUl-.30450. 15225
-.33825. 1.22 50
-.30125.10000
9".913
9567082883C6677
949131.71.5695670A2A8A.06677DODOO
5933?94510951604548575924
8933594500951064537175797
4956B53221663
*
4956
693027
ftPPBOHTHaTTON OF THE EXPONENTIAL FUNCTION
D*T»-S-T BEING USED IS 2
upturns nFfti aPFn as a - - . intinonnnnnnnnnnnnnnnnnnnnnnnn»(M
B = .1030BH00000000O0000O00DDOO0DO*Hl
ABSOLUTE ERROR CRITERION USED
A-32
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RATIONAL APFROXMATION R(K,L) ~ K = 0 , L = 2
.965607671.172<.927710865529677D+00 -.101H9181i(«628(l061!l90M<>5J512D+01 .37a7S8323Z<t65<>70<tl637678S7580+00
- .1000.00000 OOOOOOOOOGD0OO0OO0OD+O1. <nnBnnnnnnnnnnnnnnnnnnnnnffnnn*m
.53»m76I21756<*8't752399*7{i57D-01 .77720V3M178783637<t6<.39111350*00
.82
apppnyTMJTTnM .i 1 — if - i. t = n
.179533*8581285021717500992990+00
"l"is'IIoi"'?3Sns23RqMn»7Afi°3ntnii
.55780-02 2.26. 6 7
.100135102 R993 It 3 8150 fcB25355n7D+m-,313866995716'.021751025"i9[)080DtO!)
-.1000000000000000000OOOO0O0O0D+G1
.17990-02 2.75. 58
"ATTnUAL APPRnxTMATtON R IK .L I ~ K = 1 .
.9a9A6549823l'6162ftB^3ai9l.0966Q*00
.ll>71.7'i5969079B86127162i.8S7870tOO
-.1000000000 000 00 00 00 00000 000004-01
-,6821122151Q!»2063525<i897SaaOOO*aO.1onnnao0nn0ano0oDfl000nnnnnnon4.n1
-.605361755 820l>3983'»B32lt9461250*aO. 1 nonnnnannonnnnnnnnnnnnnnnnnn+ni
L = t
.33955053725653151217866923690+00
-.500 7335 6280601816<><i2/7i)323670t00.1 nnnnantinnnniinnnflflnnnnnaanaaD+ni
.<t95207867JB610081799>.0<.6561<.D-01
.178 03180919 ".".3196021.10722m 70 + 00
-.65tS7*150S20»166172*BSOH867D+00
.298501.597957372371706 ".313151.D+00
.16770-ng 2.7:..56
I
BtTTnmL APPBIUCTMATTOM BIIC.L1
-.13925Z13981023399Dl|lill68<.653a»00-.10185815266726671283972508160+01 .S2h72S32ROI1926fc7niHi2ai 795612D+0 0
-.100900000000000000000000000004-01 -.35951.179383618920830515627850+0 0• innnnnnnnnnnininnnoiinnngnooonn+iii
.3999591057<»5irB3<i906ai276ei5D+00
. 8 6
PtTTHMAL APPBn«TM»TTOM P I K . L I K a It . L
.997307720 7606 IH.JP6e?i789?T7M)+IH),l773«.7'.10708299971C82758S9<.5D*00
-.loooooooooooooooaaooooooocoootoi,33»772l.02l.<>63161J302'l0725268O+00
.'(i.l55S1927911252i.809.ii557&37D-01
,820303l*72l,5359ej69766573780Dt00-.278850201351215756599203B578D+00
.100 0000000000000000000000DODD+Ot
3.26.97
RATIONAL APPROXIMATION ROC.L) — K = 3. L
.1*355668826683 79681.0 631.770 381.0-01 -.2<>1587779<t271i>91312020909<i97D*'0 0
- . 1000 D0OO0Q 000 00 00 fO 0 0 0000 00 00*01 -.765839i*l»108350la872'.07253612D*00 -.201865930 2161(13 7«i<i22l» 16 0685 30*00-iniinnnnnnnnnnnni)niinnnnnnooonn*-ni
.131,60-03. 6 5
RATIONAL aPPROXIMftTION R(K.L) — K = g. L = 2
.10080725<>76725163gBg9<i<»17762D»01 .5006362297807882856668201*3330*00 .85a2q36939B0635M*37[)3799S325O-01.777Dai»<i75ZOa981iiia88926Mt6850-ai
-.100000003 8030 GO 00 00 0000 00 0080*01 -.7255560571831*6220521822701350*0 0 -.119Z5261'*3*593Z7528757i,0023i|D*00.innfinnnnnnnnnflnnannnnnnnnnoiin*iii
•86S2D-J*
«PPBO»THflTIOM RIK.LI — K = 1 . L = 3
• 999877681135U 1.11171691*29621360*00 .25365081.273763 nq5?D3639?l*7n3D*n n -.71.651.0903 ??l»l971.591 ?737i.2BS3n* 1J• Zl*51325318S315602l.01l.265369aD*00 -.371.65288321.551.052008909171520-01
-.iOOOOOOCOOOOOOOOOOOOOOOQOOOOD+01 >.67<t0075839221ii3it5263266383570*0 0 - . 33370 051">588138909D919'i93289D-01• ingnnnnnnnnnnnnnn(innnnnoonnnn»ni
.12i»00-Q3 3,9t_. 6 4
RATIONAL APPBOXIHATIOH R(K.L) — K = n. L = U
.10BO'.»687B33S15I.Ofll.l718aa3350*01 -.lH001'*6313263»79<.9(ia29l367»30*Cl-.1738<.18005175l.l*7751'.97386873Dt0() .361213759866590901*712371.1.1910-01
-.10008000800000000000080000000*01.575778938811325056966>9336389D*00
-.60692l*5699231l*932<*7'*18«Z67080*0 0 .5104912798<*78593850135687270D-01• ioQnnnflQOnnnnonnooonooooooQtin*ni
3.33. 8 7
RATIONAL APPROXIMATION RIK.L) — K - ». t = 1
. 999992051*88576693768706039180*00 . .a0<*30a551»?637368151323951627O»00 ,30'*'.7709216a63363132S65512ai Q±fl D
.6895S990'.136H?'*539711309Z211O-01 -.i9S6&825S1677'*10715512*8il969D*fi0
-.100000080000000000000D0000000*01.a017<.<»73S6517l.0712J'*2S793526D-01 • 5SB33lS39926581171l9'i522l)072SD*0a
-.<>380060ZS1536532't265S0S32170Q*30• 885a56t.25<.01l.379035173136760O<-n0
.18000000800000000800000000090*01
.8962D-05 5.05.70
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RATIONAL APPROXIMATION R(K,L) -t L
.57<til698<tB3>i3595li616&99Q3:i.5BD+DD
.1 .pi ftgcgQtsnaRfa.'fnfcyyT&Qi 7779faRn*n?.lM»375B5BD09<ia535BeSB3D79<>Ii0D+0D
.7025903ZZ3233't85BS053636't87lO-tll -a<t597773937li5<»8it32960355?S3SZO-0Z
-•100008000000000 00800000(00000*01 -.90590<tl05S8<i7951Z98283903256P+00 -.64764967240*5 019023371Z53<>930+00
.7567<.356S985U59S2i.a70199B573D*0 .9380S7967<t06<i03<>51582813<)2Z&p+00 .lOOOODOOOOOOOOOQDDQOOaatlODODOtOl
.5S78D-08 S.Z5.50
RATIONAL APPROXIMATION R1K,L> - - K = 3 ,
.99999999762092568499231429760+00
-.18616133650540907844490203250-01
-.iooooooaaoaoooaooaoflDgaoi>oai>D+oi
.76867667717793398254796919700+00
.5493D-08 8 .26
RATIONAL APPROXIMATION R<K,L> — K = 2>
.10000000917274786273078300110+01
-.1000000DDOOOO000S0000000OOQOD+01
•7800482749580862799019074034D+00
.89720-08 8.OS.62
RATIONAL APPROXIMATION R(K,L) — K - 1 ,
.10000009016095435266115187270+01
-.233704Z7469D50104128461965050-02
-.loaooaaooaooaoooooooaooDooooD+ai
.79035598928637463703613361310+00
.2627D-07 7.58.74
L = 4
.4310116Z00402731Z97585412235D+0 0
.1134679786589449296361342485D-0Z
. 13890997122277789311971473 8911*11 n
.94131283671364719557387254750+00
L = 5
.28753443480393714002417198260+00
-.3T313095329Z5Z6197<l9B92597470-a3
-.B9199Z3060a81S97Z31709984456D+00
,9*438971B330458S7153456556930+00
L = 6
.14381929639697650709736532100+00
.18393120231675997637174846940-03
.94713269202859761187463349840+00
•7Z305136889153345B4Z5S2343090-01
-.6270349461313*31027595*539330+00
.100000000000000000000OOOOOOGD+Ol
.24116B66275141V300269235*7Z60-Ol
-.60528360*06781311590628018060+00
.10000000000000000000000000000+01
-.85618049292*9*8483705*8744040+00
-.581513788076*9784996473873B5D+00
.10000000000000000000000000000+01
>.I
RATIONAL APPROXIMATION R(K,L» -- K = 6 , L
,100000000020*18288J77«I|18<.ZD*01 .7517015675<>3555<iaB6i|23373Z95D+00 .26925169389788336778628827360+00
.5089IB30900527011893337678B2D-01. .1755013BB68561Z0651377540971D-01
-.toooaooooaOODOOOOOOaa00000000*01 -.933 07667a 03988990564993227620*0 0 -.73965056404072712716098062770*0 0
.54428159593375538283380600450*00
.39210-09
.789997872612788 13341SZ1230dlD*00 .94636034 3564408919580480448 00*
.65
RAUONAJ. APPROXIMATION R(K.L) -- JJ_ = _
. 9999999998643815884301215340O*on .6269990313273420614646B 0223011*0 0
.30124626121657188890815971380-01-.373B0096997Zr796aZ31085SSll>9D*00
J303017573231976'.917'.701i6907D-0 2s g 6
• 1SZ193655S9'»95771.79J68035S72D-03-.?«q772fil.6Bi, 721.8853203 678<.Q760-Q2
- • I M J U pi! gojMuuLf o oooao»OBflooenp»oi-.«3Z63Z1BO«6Z%O7765781<»1I>B 7700*00
.S6133»83»5ZfclHi96Z76<»a6207niZD»l)n
-.9287132-.892350238005066884760 76S08650-01
-.7279*17B99791196q560686B^R77D*nn.255351J. 1.2 ".1631.861.6579 SOOtn'tDt CIO
.1000000000000 0000000000000000*01
~.m*o-W 9~77i
RATIONAL APPROXIMATION R IK .L ) — K - >t, L = I|
.S01991<t8993076463aOZaol8S2560«'0 0
.10608631211.3111.2926575280 3660*00 -.116757a55608ZZ33909Z<>ZZl>S5a6a-01
. 108077798035973 929968 593961. ID* u 0
.gqBnnBgnqys5ft73fl2itiRi.fi7ijf.fl2nt.in•57386603202730353803387682130-0 3
-.10000000000000000000000000000*01 -.9Z<t8ie9Z55<.3767597<t 7215575930*0 0 .7150Z382521623Z7709l>5a9l|S96tlO*0 0.777H331fc5q.TI.q70 73362l.93lt 75520 »nn
.57727581996e2690757813i.3137aD*00
.iooBojoooaonnnnnonnnfloonononD*oi. 807ZZ009559<>919 058598 0098672D*0 0 .9511050991.753 53 61.9651561.1.3320*0 0 l
00.15380-119 9-81
.Mi
RATIONAL APPROXIMATION RIK.LI -- K 3 . L
.9999999998?!»&753629997711520*00 • 3767316557977366890 qa2D7563l.0*Q0 . 5409661.12931.3 2a6B16a2Q06qqll.n-ni
.302071.566575863581.33765279860-02-.Z93767l6B07a300339Z7569055aaO-01
-.62326831.1.89531510008950198150*00• Z9056597Zli.if99't9'l'.lB1185Z9(.90-Og
.1773&l>97937Z2570731>5l.u51l«8b5O*00.lltl99Z9Z0 0<.9851ZZ3Z9815625e'.D-03
-.iaOOOOOOOOODflOOOBOPOOOOBOOOOO*01-.3eB96577566328l.7333755935D650*00
.5921.51.77983020811131.1) 31.23 7620*00
-.920S358«.a87S979SS9<H697761310*00 - . 701217187l.Z96839576Z7l6ai.l.590*nn-.39061. l.73S2237976'.20011370a39D-01
.Blfc 860365 33Z2316'»Z91'>l.31fc75nO*0 0.30000196593 69386&269571I.1302D+00
•10000000800000000000000000000*01
.18970-09 9.7Z. 5 *
RATIONAL APPROXIMATION RIK.UI — K Z, L = 6
.10000000003698513290al.027056D*01*00
.88090 28^551033 8133Z6B799b60D-0Z
.2512750 85 0331.710272 50 71111.01D+0 0
.26676B151635307827t7i.99i.09120*0 0-.87060268877156522181.059171330-03
.1801.3222826839061.161.961761.5 80-0:
.5907Z375303a6aiOZ67Z7Zl.7361t30-01 _
.1.682215101.1.251.22631.0660877 010-0".
-,100DOODOOOOOOOOOODOOOODOOCOOD*01- .366B3a917787in729172793319fc70*BO
7 3 7• 6070<i067903089987i,75211673900*00*oi
- . 9 1 5 9 6 7 2 2 0 81087S6029636207'ta'.D*DO-,139gl39' .Z<.3n7Z76' .2ani763ZS51O-01
5 2 7 5 7 6 5 0 0 0.8220821.5861.712177807547865900+00
.68658701.99867 8179^800a57276l.D•aO• 3?lSOSOI.qiqSB6a8266021173517O»n0
5 6 1 7 3 6 7.955061.73268795222900817842960*00
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//CONTROL//* \ o T fi I. -.
APPROXIMATION OF THE EXPONENTIAL FUNCTION .
_O*TATS£T BEING US££LJS_-_2
BOUNDS OECLARjjp AS A s -.lOOOOOOOOOOOOOOOOOOOOOOOOOOOO*!!!
. _ £ = .10000081100811000000 0000 0000000*01
ABSOLUTE ERR98
REP0B1 FOR APPROMHATION Rl H, B) PRECISION I-LOEIO(H)) « 3 .26
RECORD WUHBER OH gOAT IS a n i l
ESIiSMEBJtOSS IH PRECISION DUE TO COHDITIONING IS _«a O1SATS,
PflOPSlPB2PB3
1<f
I)09SJ_0)
.10000
.99730
.1.9883
.17734
900935*53
•'«'»155 5
//CONTROL//~~L 3 1 ~ " " ~
nF THF FyPHNFHTTAI FIIHCTTOH . . _. _. ..
JS 2. _ -
a = - . i mnnnnnnnnnnonnnnnnnnnnnnnnn-*.ni
a = .inflnnnnnnnononnnnDiiDonndaaiinnt-Di
RBSQLUTE ERROR CRITERION USEO
REPORT FOR APPROXIMATION »( ?, ?1 PRECISIOM l-l nr.ll) IM) I s h.llfi
HIIMBFB OM gnAT.IS 2XLU-
ED LOSS TN PRECISIOH DUE TO COMnlTIOWING IS .It OISITS
pnqPDiP02
aoo001aoz
iiii<
?)i)it2)1)1)
. ii'aE|<)
.65l.5lt.11045.12868
- . 6 3 1 4 6.10000
4320506138720000
REPORT FOR APPROXIHATION R< 2, 3) PRECISION (-LQS101HII = 5.37
RECORD NUH8ER OM R.DAT IS 2019
ESTIHAT6D LOSS IH PRECISION DUE TO COHDmONIN6~IS~ .5 DIGITS
POO t 2) - .6W79 37632EM. X i ) -.26001 72896P02 I 1) -.32832 7815
_OQJL I 2) - . 61 .279 lt8S212I 2) .38276 31816
O03 ( 1) .10000 0000
REPORT FOR APPROXIMATION R( 3, 3) PRECISION (-LOSIO(H)I = 6.81
RECORD NUMBER ON 3DAT IS 2025
ESTIMATED LOSS IN PRECISION DUE TO CONDITIONING IS .>> DIGITS
POO£01POZpu
3) -.12588 7&SJ* 53i\ -«.6313/. 9JJTS1 i2» - . 1 2 7 9 0 <»<t<tS5 2II ,-.11173»
900 ( S> -.1ZSB8 78370 87801 I 2) .62099 817» 9Q.OZ ( 2) -.12336 70686 6993 < 1) ,100 00 09001)
FOR APPROXIMATION _R< 3, %
RECORP NUMBER OH ROAT IS 2033
PRECISION (-L0S10O1I) S..Z6
_ ESTIMATED LOSS. IMPRECISION DUE IÇLÇflNfilTJEOJUNS IS . . ,5 . OJ6ITS_.
PBO I 3> .88130 Sqn62 l>75P01P02P03
Q01an?
I 3)
1)_J1.
.37905 30873 0659JA3J» "til39597 6
.»ai3n3) - . 501*5 2787» S<i231 . 1 2 t S ? gfiQlifl HRQ
90S ( 21 -.16M16 50857 67. I U_jiM!UL.flJUliUL 0
REPORT FOR APPROXIMATION R! ». ») BREWSI8N <-t.OSin<HM
RECORP WUMBER ON ROAT IS 20»!
ESTIHATFO LOSS tN PRECISION OUf TO CONOITTgNTNG IS OISITS
h) .17025 66983 892613) .87475
OKUUL37960 3385
P03POfc
.210.18
.10570061.32 22931622 21
Q00UO1
".I .171(2531 -.86781
66982 867573185U 1603
Q02 t 31 .18*86
ana (250>i8 7%77S99B< 197
I 1) .10000 00000 00
-' XREPORT FOR APPROXIMATION Rl <tf 5)
RECORD NUMBER ON ROAT IS 2051
PRECISION I-LOS10«M)I = 11.37
ESTIMATED LOSS IN PRECISION DUE TO CONDITIONING IS .5 DIGITS
POOP mPO 2PnlPOkqnnQO1 <
403 <Qnb i005 i
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41*i1)5)If)bl
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-.15686-.KqqiïÇ-.13167
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56085
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753
11292<>37Uktt81S7171?000
22
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REPORT FOR APPROXIMATION Rl 5, 5) PRECISION (-LOS10CMI) s 13.01
RECORD NUMBER ON ROAT IS 2061
ESTIMATED LOSS IN PRECISION DUE TO CONDITIONING IS .4 DIGITS
POOPB1PD2pits
51 -.31148 39046 29302 83651 —1CJ613 Sq>51 ES376 26641 -.34798 10802 99573 32
PD4 2) -.31286 66347 2961911 -.1H46II 52<t32 0621.
Q00JUU_002oa-i
5) -.31148 39046 29277 4235» .1SS34 79194 437H8 5304» -.34404 07516 04737 03
n«04005
2> -.30351 26651 09672
-11 .lnnoo ooooo oooo
//CONTROL// 8 3 2 2 **'
= ========== = = == = ====== =
APPROXIMATION OF THE EXPONENTIAL FUNCTION
OATA-SET BEING USED IS i
BOUNDS OECLARFO >s > = - . lDannmnnumpnmioiin nnnn nnnannnm 1
a =HEIGHT FUNCTION 5UPPLIE0 IS USED
//CONTROL// A 4 0
B-TTflM-i APPPniCTHaTTpiJ B€K,I.I — K - U, I = ___
•san+nn.17-*8623<l93t7B0875<> 797365 374O+O0 .3996291659B16S50298590927998D-01
-.lOOOOOOin) 0000 00 DODO OD 000 ODODD+Ql - .8567908 030929 0292207646287850+00 -.44741084170 554926096<.7<.9SZ56O+DO
. 8 7
B-TTOMtL -PPga»IM»TrilW RIK.L> — K =
--<fiflnnnnnniinnnnnnnnnnnnnnnflnnn4.nl .1 nnnnnnnnnnnnnnnnnnnnnnnnnnnn*ni
-76l_DtOO _ _ _ .o.ao
g , | 1 —- y s ( i .
»»ERRnR»> M»»tH- OF ERROR RIIBtfE DECEWERtTF
BEST-BT »TTEMPTEtl fSIEWT BESET!
RATIONAL APPROXIMATION RCKiL) - - K = 0, L = 1
• • E R R O R " HAXIH* OF ERROR CURVE DEGENERATE
BiTTIUIlL aPPgfUTM-TTIItJ D l t . l I — K s 1 . 1 = 1
.1000000000000000000OD0D0DOO0O»01
RATIONAL APPROXIHATION RCK.L1 — K = 1 . L = 2
,998332959'»69ii3979385'l73b83200tOO .325«.337663<i3569D8a509Z309536O»u0 -.67223332997D39181767957tS6i30*00
-.<t»BiBil_B_rinnn_nnfliinnnnni<iin*ni --_7a7_<_87-i i ig3gaBqns7nU3^n»nn.7*30B23l|2266»'»9593S9<t7<i23'i76D+00 .100 00000000000000000000000000*01
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.15330-09 9.81
RATIONAL APPROXIMATION R(K,L> — K - <., U = 5
• 99999999999571f725785702 50573 D+0 0 .lll|<. 399137 58 0932961.0 38311851.30+0 0 .8327*3 0569525173B659521B3 3M.D-01
. 13887 516789B9617593-383771.303D+00 -•19B 2395 0661|5327129070 657601.10-01 .16<,835«.6S2b 1(13951731.331628790-02
-•innnnnnnnonnnnomninnnoGODnooo+oi -.9502089732532081231. <.SI.6735a2D<-nO-.300<i62972a33<>58165Biia<t35B900D')'0 0
- .anan i 8705601113 ifli77aii?0G77n«-nn.10125<>90929345190B2370i><t76B3D-01
,9521371't31139i1b691918D180795D+0 0 .lOOOOOOOOOOOOOOOOOOOOgODOOOOD+Ol
.1.271.0-11 11.37
RATIONAL APPROXIMATION RIKiLI — K 5, L =
.100000000000000OOD0000000627D+D1 .«»999999999991>Sl»7&Z7978060295D4'00 . Ill0831.23229B7660 8356160111.20+0 0
-.K999999999991.1.96129591771716D+0I1 .111083'.23229859033l.368ai.7663D*0 0- . 32B39R9aa 69071B331 B7tH6l.n HI 70-01.
- . 13875 01.1.958 91.89565676969792 90-D1
-.innnooniiininiinnniiniinnnnnonniinn*oi-.655't61'.775023a2205097965296aD+00 -.l*160072516651032189838652573O*0 0
,8i.l577S9i.259065953Di.38H50e56D+00 .95955623707212369272820301.270+0 0
-.11.2559580899608889719351.91330+00.6551.611.77^039283195 0959^31 fin<-.in.100000000 0000 0000000000000000*01
.97080-13 13.01
1X3
//CONTROL// K <<
APPROXIMATION OF THE EXPONENTIAL FUNCTION
DATA-SET BEING USED IS 3
B O U N D S D E C L A R E D AS A - -.loooonoooDOOoooonooonooonoooo+oi
3 - .iDQQQODQuaasDoaooaaaoaooooDoo+01HEIGHT FUNCTION SUPPLIED IS USED
REPORT FOR APPROXIMATION R( •» t 0) PRECISION (rLOG10(M)) =. 3.3j!
RECORD NUMBER ON RDAT IS 3011
ESTIMATED LOSS IN PRECISION OUE TO CONDITIONING IS .9 DIGITS
POOl>01POZP03POlt
<(
ctt
010)0)0)
-1)
.99962
.99793
.50299
.176 <.fl
.39962
790873a 656 2 39 ?
========= = ==== = == = = =====:=: = :==//CONTROL// L 3 1
APPgnXIMATTnN OF THF F«PnHFHTT&l FIIMCTTnw
HATt-SFT HFTMC USFn IS 3
nFP. --innnnnnnnnnnnnnnnnnnnnnnnnnnn»ni
a = .lomnnnniflDnnnnDnmnimninnnnnnfuni
HEIGHT FUNCTION SUPPLIED IS USED
HfPllBT FOE UPPBOKTMttTION HI ?r ?> PRFHTqlnH l-lfir. lG(M)) = u.nft
BFp.nan HIIMHFP nu anflT
FSTTM»TFI1 LOSS TM PgFRTSTnH HIIF Tn RnMnTTTnMTMC T?; .It OIGItS
P01
QOOo m002
Cf
c
It11
21
11
.61253
.1nnfln
.12256
.10000
9500nnnn02623
OBOO
REPORT FOR APPROXIMATION RC 2 , 31 PRECISION C-LOG10CHI) = S.36
RECORD NUMBER ON ROAT IS 3019
ESTIMATED LOSS IN PRECISION OUE TO CONOITIONING IS .5 DIGITS
P00
P02annQ01an?
21 - . 6 1 1 7 2 015>>6 S
II - .30213 86908- .6117? ZT6T6 2
2) .36740 73210 373670
903 II .10000 00000
, REPORT FOR APPROXIMATION R« 3, 31
RECORD NUMBER ON RDAT IS 3025
PRECISION (-LOGIO(NI) = 6.81
ESTIMATED LOSS IN PRECISION OUE TO CONDITIONING IS .•> DIGITS
POOPO1P0 2f>03QOO0.01QO2
S)1 2)
2)1 )
212)
- .12151- .60757- .12125- .10000- .12151.50757
-.12125
1.75 85313711736600000475553137117366
1.6!696 6
n1.6167S5
REPORT FOR APPROXIMATION R( 3, "•>
RECOBP NUMBER OM RDHT TS 31133
PRECISION (-LOSlll(t | l> = 8.26
ESTIMATED LOSS IN PRECISION DUE TO CO.NQI.TIONING IS , 5 DIGITS
PODP01P02P03anaooi00 2403
; 5)1 3)1 2)( 1)
J)3)3)2)1)
.85055
.361.1.1
.60621
.".0156
.85055-.".6611.
.1211.8-•16158
• tSiPB
6350S15232Q«<|656<»9<i63555"•82557S4SI.7 83 65oogoo
638Hi . _ . . . . . . _ ._g649 _. ...5756 9 5070
I
REPORT FOR APPROXIMATION R( <>, •») _P_RgCISIPN C-IOS10(Ml 1 .=
SECP8D NUMBER ON RDAT IS _JJ»t-. . . . . . . . -_
ESTIMATED LOSS IN PRECISION OUF TO CONDITIONING IS .» PltlTS
P0BP01P02PD3PO<tQOOOfll402003QO<t
[ *>>
( 31[ 3)( 2)1 1)I <•>
t 3)2)11
t16950.81.751..18150.20125ilflMIL.16950
•18150- . 20125
.10000
9306265301797 681011.7ODOOO9306265X0179268
oaaoo
51*77^ 705980
91.32Q215<t77l< 7
6763dqi73ooa
REPORT FOR APPROXIMATION Ri <•, 5)
RECORD HUMiER ONROAFrS ~30~51~~
PREKISION (-tOS10«M)I = 1 1 . 3 7
ESTIMATED LOSS IN PRECISION DUE TO CONDITIONING IS .5 QIGITS
POOPi l lP02P0 3PO4anaQD1aazQO3 1a n * iQ05 <
5)i>)4>31It5)•ll413)2111
-.15258-.A78I18-.12706-..i?n«D-.50125-.15758
.84776- .71 (90\3024S~.?m«l
.10000
56917quqa346754qnn?i8093556917
Jfi2£fi-51226•;33kqaoooo
819705386?2227391ifiq59582S14629135290 50122576fet>000
7625
REPORT FOR APPROXIMATION R( 5 , 51 PRECISION (-LOC10CKM 13.01
RECORD NJMBER ON ROOT IS 30«l
ESTIMATED LOSS IN PRECISION DUE TO CONDITIONING IS .<• DIGITS
POOPHIPO 2PnxPO*p a * •
QUOamQ02
O0<i005
[ 5)S)
1 <*i31
I 2>1)
[ 5)r 51[ *>)
2)11
-.30450-.15225-.33825-.fa775D-.30125- . inn no-.30450
-1S775-.33825
-.30125-moon
9491347456956708788306677nnnnn949134745695670
06677JUUUUL
893359451095160454S575924anaK389335<i45na95106
75797naiian
49568532216638
49561447699n?7
tn
. J- .-
A-54
i i
LISTING OF WALSH ARRAY
r.TTnN
PITH-SET BEING U
BOUNDS QEnLflRfn AS a - -.innnDnnGOQODDnoooooQDOOOonnnn»ni
B = •inQnnnannaonnnDnnononnnnnnnnn»ni
RELATIVE ERROR CRITERION USED
0.00
.63
i.4Q.83 .S3
2.30 2.76 ••• 2.30,K7 .57 .B7
3.30 3.89 It.06 3.89 3.30.87 .65 .1)3 .65 .87
4.38 5.07 5.36 5.36 5.07• fl7 .tig .<;? ••;? .fig
.72 .58 .•»! .586.29.72
7.56 8.04 8.26 8.26 8.0.ft?
II
9 10
q
. 8 3
.42 9.72 9.81 9.72 9.1.2
.65 .5". .43 .54 .65
11.19 11.37 11.37 11.19.58 .48 .48 .58
10 12.69 12.93 13.01 12.93 12.69.61 .52 .43 .52
NUHBER OF ENTRIES NOM PRESENT IS 45
SI 1N3S33M MON S3IB1N3 dO
T 9 ' I V T 9 -ZS' I*6"ZT B/TT
"S?" ST" SV ?5» ZT"6 f T T / f î l it*TT OZ'TT «H'OT
• <lS' <I<I' « S ' S9"5 9 ' «lS ; s ' S9ZT*5 f i t
sz*e 9Z*9 so-e es*/
es* »»• gs* ?/•S9*9 19^9 69^9 Tî '9
SO'S 9 t ' S it'S 80*S
8 5 * 9 S * 9 8 'il'Z 8/*2 SS'Z
SS' «9"
TtJ+UOOuODDDDODDODDDDOOOUOODOOQul" = H
Ta-»otnraooocooDDODfîoooooiiooDDDaD't>- = » s» asuvisso sonnée
2 SI O3SI) SNI36 13S-»l*0
NOliJNniIVTiN3N0d«3 3HI JD NOIIVNIXOBfldV
HS1VM JO
LISTING OF M»'.;H SRRAY
nr THF fnpoNFMTtai FUMCTTCM
D*T«-SET BEING-USED IS 3_
anilMni ncn teen a< a = -.innnnniinnnnnnnnODnDOniinnnnniin»iii
-innnnnnnnnnnnnnnnnnflnniinnnnnn»ni
WFISHT FUHCTIOM SUPPLIgn IS
• 120.00
1.68
3
5
2.76
.1.3
5.36
. 87
6.26
.Hi
11.37
NUMBER OF ENTRIES NOW PRESENT IS 12
o-
13.01.<t3
19/01/73 AECL SCOPE 3.2 VERSION C 11/01/73
ll.'i'i.09.REMEZOO,BZ35-JNBtCN1100008.
11.44.10.CYCLE 10! REMES2ALL.hk.ia.FH.E HAS BEEN11.44.10.FTN.IL+kh^lZ. _ .JU9CP SECOHOS-JIQMP.IIJUIQN TIME11.44.16.LOAD(LGO)
11.44.26.EXECUTE{RE*£S2,LC=2000D>ll.hg.2P.EST. LOAD FL REnUIBEOa 31057 1071.5216111.45.24. ••••R.DAT • • • •n.fcg.?!.. • • • • T O T A I msieagFt • •••••(.i.n wnans11.".5.21.. ""TOTAL ACCESSES ••••••••0 THIS RUN
ll.*5.2%. ""TOTAL REPLACES »»»»»»»»0 THIS RUM_ n-ug.ys. » » » » T O T " HFI FTFC T«»»»T»»n THIS RUN1L.4S.ZS. **** NUMBER OF ENTRIES AVAILABLE IN11 .tg.gfi.paTWARY rmnb» »»»»>»»» fi11.".6.03.ROLLOUT COMPLETED. (FL 67500)
11.1.8.20.ROLLOUT COMPLETEO. IFL B7500)JJ^M.W.ROLLXN iOflPLEJXD.^I l . fc8.55. ••••RDAT • • » •i i . t i i . g ' ; . »»»»Tnrai nisieagFa »»»«* I .7^R ungns11.48.55. ••••TOTAL ACCESSES ••••••185 THIS RUNi i -hH.si j - • • • • rnra i TM«:FPT«: » > » » » » I I H ; THTS BUM11.".8.55. ••••TOTAL REPLACES • • • • • • • • ! THIS RUNil . l i f l .S ' i . ••••TflTAl OELFTFS ••• • •»««n THIT. BUM11.%S.55. • • • • NJHBER OF ENTRIES AVAILABLE IN11 .m.pS.PRTHABV IMIIEX »»»»»»T»gg11.49.09. »»»»RDAT • » • •1l.hq.nq. »«*»TnTAI niSKARFA »»»*«I«73R MOHnS11.49.09. •<-"TOTAL ACCESSES ••••••105 THIS RUN11.49*JL9. »»tf«TOTAL INSERTS •••••••«n THIS SUN11.49.09. ••••TOTAL REPLACES ••••••••0 THIS RUN
. 11.49.09. »»»»TOTAL DELETES »«»»»»»»p THIS RUN11.49.09. ••*• NUMBER OF ENTRIES AVAILABLE IN11-41.nq.PBTMABT INnFK •»»»»»»«2?11.49.19. ••••ROAT »•••11.49.19. ••••TOTAL DISKAREA »»»»»47I6 WORDS11.49.19. ••••TOTAL ACCESSES ••••••••0 THIS RUN11.49.20. »»»»TOTAL INSERTS ••«»»»»»Q THIS RUN11.49.20. ••••TOTAL REPLACES ••••••••g THIS RUN11.l.q.2H. »»»*TnTAl OELgTFS ••••••T»p THIS RUN11.49.20. •••• NJHBER OF ENTRIES AVAILABLE IHtl.49.211.PRIMARY INDEX *»»»»»»»2211.49.22.STOP11.49.27.CP 049.599 SEC.11.49.S7.PP 133.745 SEC.11-I.Q-77.HU 0(14.818 HHS.11.49.27.
.PUNCHED £A_RO_S
BFMF787I
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