%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
TABLES FOR WAVELET FILTERS ADAPTED TO LIFE ON AN INTERVAL.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This file contains two sets of tables, both for wavelets adapted
to the interval. The resulting wavelet bases on say [0,1] have
the same number of vanishing moments (even as the edges) as the
unperturbed interior wavelets.
The first set of tables corresponds to a construction which was
discovered independently by Albert Cohen, Ingrid Daubechies &
Pierre Vial, by Bjorn Jawerth, and by Pierre-Gilles Lemarie .
A first announcement of the construction appeared in
A. Cohen, I. Daubechies, B. Jawerth and P. Vial, "Multiresolution
analysis, wavelets and fast algorithms on the interval",
Comptes Rendus, Acad. Sc. Paris 316 (serie 1) (1992) pp. 417-421.
The full details of the construction have appeared in "Wavelets on the
interval and fast wavelet transforms", by Albert Cohen, Ingrid Daubechies
and Pierre Vial, in the first issue of Journal for Applied and Computational
Harmonic Analysis, volume 1, 1993 (pp. 54-81).
The second set of tables corresponds to an older construction of
wavelet bases adapted to the interval, due to Yves Meyer,
"Ondelettes sur l'intervalle", Revista Matematica Iberoamericana,
7 (1992) p. 115-133. The original paper of Y. Meyer did not give any
tables corresponding to his construction; they are provided here.
For a discussion of some advantages of the new construction over
this first construction by Y. Mayer, see the J. ACHA paper mentioned above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
TABLES FOR WAVELETS ON THE INTERVAL AS IN THE JOINT WORK
OF ALBERT COHEN, INGRID DAUBECHIES AND PIERRE VIAL.
(Tables computed by Albert Cohen)
***************************************************************
What follows are tables for a special construction of wavelets
on the interval.
The number of scaling functions and the number of wavelets
at a resolution level j are equal in this construction
(and both equal to 2^j).
The tables are carried out for different values of N (N = 2 to 8),
which is the number of vanishing moments of the wavelets.
For given N, we need to adapt the N outermost wavelets and
scaling functions, both on the right and the left side of the
interval; the 2^j-2N "interior" functions are untouched and
are therefore the same as for the construction on the whole line.
These are therefore associated with a pair of exact reconstruction
filters. We have chosen the pair with length 2N (shortest length
for which N vanishing moments are possible), and with phase closest
to linear phase (exact linear phase is excluded, because these
filters are orthonormal).
At each edge we have computed the adapted filters corresponding to
the adapted functions. Their lengths go from N+1 (for the one closest
to the edge) to 3N-1 (closest to the interior), with increments
of 2. The adapted filters are different for the left and right edges.
The following illustrates how these adapted filters should be used.
Suppose the data are represented by the following sequence of stars:
* * * * * * * * * * ....
We only consider the left edge here. The approximations at the
next coarser scale will be obtained by applying low pass filters, and
decimating, as usual. We represent the result as a sparser sequence
of stars:
original seq. : * * * * * * * * * ...
coarser seq. : * * * * .....
In the case N=2, the interior filters have 4 taps, and we have 2
adapted filters for the edge, one with 3 and one with 5 taps.
What this means is that the leftmost of the coarser sequence points is
obtained by weighting the leftmost 3 points of the finer sequence with
the 3-tap adapted filter (the first coefficient weighs the leftmost fine
scale point, the 2nd coefficient the 2nd point, etc.); the second leftmost
of the coarser sequence is similarly obtained by applying the 5 tap adapted
filter to the 5 leftmost fine scale points. From then on, i.e.
for the third leftmost coarse seq. point on (and until we get near the
right edge, where the right coefficients take over) we use the "interior
filters". The third coarse point e.g. is obtained by acting with the standard
4 tap filter on the fine scale points directly above it ( 2 to the right
and 2 to the left).
For larger N, one has to use the adapted tables for the N leftmost
coarse points; from the (N+1)th, the interior filters take over.
For any N, these tables can only be used if the data have at least
2N points; for less points the 2 edges interfere and these adapted
tables are not sufficient. The whole procedure can of course be
repeated, to lead to a decomposition in many scales, but it has to
be stopped when the coarsest scale has strictly less than 2N points.
(At this point, other adapted filters should take over; in any case,
if we want the scaling functions at every level to generate all
polynomials of degree less than or equal to N-1, then we can at most
go one step further ...)
The high pass filters work of course in exactly the same way.
On the infinite line, the fact that the wavelets have N vanishing
moments is expressed by the fact that
... 1 1 1 1 1 1 1 1 ...
is invariant under the lowpass filtering + decimation operation,
while ... 1 2 3 4 5 6 .... gets mapped into 1/2 times itself +
multiple of .. 1 1 1 1 1 1 .. , etc. (similar statements for
higher order polynomials). On the interval, it is not true,
even though the wavelets have vanishing moments, that the
sequence 1 1 1 ... 1 1 1 gets mapped, by the lowpass+decimation
operations to a sequence 1 1 1 .. 1 1 1 of half the original
length. (This is because the "adapted" low pass functions
don't have integral 1.) Instead, there exist a1, a2, .. aN
so that a1 a2 .. aN 1 1 1 1 1 ... gets mapped to a similar
sequence (at the other end, we need to end with properly chosen
bN, .. b2, b1 of course). This sequence a1 a2 .. aN 1 1 1 1 ...
is therefore the sequence which maps to an all-zero sequence
under the high-pass+decimation operation. In some applications
(image analysis?) one would like the sequence 1 1 1 1 ...
to have this property, however. To achieve this, one has to
precondition the data. Before starting the decomposition,
the N extreme values at the left have to be multiplied
by a lower triangular matrix (the upper left corner triangle is zero), which
will map the column vector 1 1 .. 1 to a1 a2 .. aN (and which will
also do the appropriate thing with the higher order polynomials).
See below for the detailed preconditioning scheme.
This matrix is also given in the tables, as well as its inverse
(needed after reconstruction), which is also a lower triangular matrix.
Organization of the tables:
For every N we have
list of the lowpass filter coefficients used on the
interior.
(BEWARE: these should be normalized so that their sum
equals sqrt(2). In some cases - which will be indicated -
it isn't, and the normalization should be adapted to be
in accordance with the "edge" coefficients; the "edge"
coefficients are always normalized in L^2-sense, i.e.
the sum of the squares of every group is 1).
two columns with the lowpass adapted coefficients at
the left, highpass coefficients at the right. there
are N different filters; the shortest corresponds
with the outermost point, the longest with the innermost
of the adapted edge filters.
two columns with N blocks, with N entries in the every
block. These give the entries of
the preconditioning matrix described above: left column
for the matrix to apply to the data, right column for
the inverse, to use at the end of the reconstruction
(each block represents a column of the matrix in question).
THE WHOLE THING IS GIVEN FOR THE LEFT EDGE AND THEN REPEATED
FOR THE RIGHT EDGE.
AT THE RIGHT EDGE :
-The adapted filter coefficients have been obtained from the same program as
the results at the left edge, except that the "interior" filter
coefficients were listed in the opposite order. The filtering operation should
thus be done symmetrically : in the case N=2, for example, the rightmost of the coarser sequence points is obtained by weighting the rightmost 3 points of
the finer sequence with the 3-tap adapted filter (the first coeff. weighs the
rightmost fine scale point, the 2nd coeff. the 2nd point, etc.); the second
rightmost of the coarser sequence is similarly obtained by applying the 5 tap
adapted filter to the 5 rightmost fine scale points. From then on, i.e.
for the third rightmost coarse seq. point on (and until we get near the
left edge, where the left coeff. take over) we use the "interior
filters".
-The preconditioning matrices are now upper triangular and their application
does not require to "symmetrize" the data, i.e. to consider the rightmost point
as the first point. The following scheme (corresponding to N=2) shows that
the extreme points should be consider as standard column vectors on which
the matrices are applied :
Prec. Matrix Original Data Preconditioned Data
Left | A 0 | a Aa
| B C | b Ba+Cb
c c
d d
e e
f f
| D E | g Dg+Ef
right | 0 F | h Fh
The inverse operation (after reconstruction) works in the same way.
************************************************************************
The interior coefficients in these tables were computed with Macsyma.
The edge coefficients were then computed, using these interior filter
coefficients, with a Fortran program using double precision.
************************************************************************
TABLES AT LEFT EDGE.
%%%%%%%%%%%%%%%%%%%%%%%%%
N=2
lowpass coefficients in the interior
0.482962913145
0.836516303738
0.224143868042
-0.129409522551
coefficients for the edge filters:
0.6033325119E+00 -0.7965435169E+00
0.6908955318E+00 0.5463927140E+00
-0.3983129977E+00 -0.2587922483E+00
0.3751746045E-01 0.1003722456E-01
0.4573276599E+00 0.1223510431E+00
0.8500881025E+00 0.2274281117E+00
0.2238203570E+00 -0.8366029212E+00
-0.1292227434E+00 0.4830129218E+00
entries for the triangular preconditioning matrix and its inverse:
0.3248940489E+00 0.3077926491E+01
0.3715801512E-01 -0.1142045672E+00
0.0000000000E+00 0.0000000000E+00
0.1001445405E+01 0.9985566812E+00
N=3
lowpass coefficients for the interior:
.332670552950
.806891509311
.459877502118
-.135011020010
-.085441273882
.035226291882
coefficients for the edge filters:
0.3888997639E+00 0.5837809695E+00
-0.8820782813E-01 0.7936188402E+00
-0.8478413085E+00 0.1609551773E+00
0.3494874367E+00 -0.5884171124E-01
-0.6211483178E+00 -0.3493401824E+00
0.5225273932E+00 0.2989205572E+00
-0.2000080031E+00 -0.3283013439E+00
0.3378673486E+00 -0.3322637132E+00
-0.3997707705E+00 0.6982497200E+00
0.1648201297E+00 -0.2878789995E+00
-0.9587863831E-02 0.1015059001E-02
0.3712255319E-03 -0.3930133175E-04
0.3260097101E+00 -0.3451437111E-01
0.8016481645E+00 -0.8486981030E-01
0.4720552620E+00 0.1337307023E+00
-0.1400420809E+00 0.4604064538E+00
-0.8542510010E-01 -0.8068932216E+00
0.3521962365E-01 0.3326712589E+00
entries for the triangular preconditioning matrix described above,
and its inverse:
0.1007941579E+00 0.9921209927E+01
-0.5929310244E+00 0.2752403722E+02
-0.1509649764E-01 -0.6946796955E+00
0.0000000000E+00 0.0000000000E+00
0.2137256653E+00 0.4678895250E+01
0.3068542369E-01 -0.1435467050E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1000189333E+01 0.9998107029E+00
N=4
coefficients for the interior lowpass filter:
(beware: these coefficients are normalized so that their sum is 2)
0.045570345896
- 0.0178247014417
- 0.140317624179
0.421234534204
1.13665824341
0.703739068656
- 0.0419109651251
- 0.107148901418
coefficients for the edge filters:
0.9097539258E+00 -0.7573970762E-01
0.4041658894E+00 0.3254391718E+00
0.8904031866E-01 -0.6843490805E+00
-0.1198419201E-01 0.6200442107E+00
-0.3042908414E-01 -0.1885851398E+00
-0.2728514077E+00 0.1665959592E+00
0.5090815232E+00 -0.4847843089E+00
0.6236424434E+00 0.3564635425E+00
0.4628400863E+00 0.4839896156E+00
0.2467476417E+00 -0.6057543651E+00
-0.1766953329E-01 0.3451833289E-01
-0.4517364549E-01 0.8824901639E-01
0.1261179286E+00 0.2082535326E+00
-0.2308557268E+00 -0.4018227932E+00
-0.5279923525E-01 -0.6872148762E-01
0.2192651713E+00 0.3302135113E+00
0.4634807211E+00 0.5580212962E+00
0.7001197140E+00 -0.5994974134E+00
0.4120325790E+00 -0.6909199198E-01
-0.2622276250E-01 0.2785356997E-01
-0.6704069413E-01 0.7120999037E-01
-0.2907980427E-01 0.6548500701E-01
0.5992807229E-01 -0.1349524295E+00
0.6176427778E-02 -0.1390873929E-01
-0.4021099904E-01 0.9055141946E-01
-0.3952587013E-01 0.8900857304E-01
-0.5259906257E-01 0.3733444476E+00
0.3289494480E+00 -0.8404653708E+00
0.7966378967E+00 0.3156849362E+00
0.4901130336E+00 0.1202976509E+00
-0.2943287768E-01 -0.1307020280E-01
-0.7524762313E-01 -0.3341507090E-01
entries for the triangular preconditioning matrix and its inverse:
0.2489911114E+01 0.4016207624E+00
-0.2752988536E+01 0.6592239446E+00
0.1687841447E+01 -0.1870967456E+00
-0.4022221173E+00 0.3352374610E-01
0.0000000000E+00 0.0000000000E+00
0.1677210550E+01 0.5962280646E+00
-0.7075375437E+00 0.3732739492E+00
0.1763544288E+00 -0.8158414568E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1130145142E+01 0.8848420994E+00
-0.6162121550E-01 0.5415219932E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1006885156E+01 0.9931619247E+00
N=5
coefficients for the interior lowpass filter:
(beware: these are normalized so that their sum is 2)
0.0276321529578
- 0.0298424998687
- 0.247951362613
0.0234789231361
0.89658164838
1.02305296689
0.281990696854
- 0.0553441861166
0.0417468644215
0.0386547959548
coefficients for the edge filters:
0.9302490657E+00 0.2803062751E-01
0.3488878121E+00 -0.1899596351E+00
0.1098578445E+00 0.5126446690E+00
0.2701025958E-01 -0.6992703507E+00
0.7897329608E-02 0.4475860840E+00
0.7300852033E-02 -0.1049415950E+00
-0.3099501188E+00 -0.6736388320E-01
0.5987214008E+00 0.3342473230E+00
0.6208411495E+00 -0.5086311754E+00
0.3720164550E+00 0.1104945888E-01
0.1451060379E+00 0.6367318477E+00
0.6263262144E-02 -0.4656214411E+00
0.1687072021E-01 0.3800200446E-01
0.1562115518E-01 0.3518730685E-01
0.1138401479E+00 0.1327317986E+00
-0.3641831811E+00 -0.4544562923E+00
0.1228916167E-01 0.2604140629E+00
0.4117590210E+00 0.4050120843E+00
0.5954973448E+00 -0.8108043041E-02
0.5457944280E+00 -0.6031278756E+00
0.1756416185E+00 -0.3803908839E-01
-0.1751671276E-01 0.4149398330E+00
0.2376581050E-01 -0.5938666471E-01
0.2200554624E-01 -0.5498806769E-01
-0.4098816981E-01 -0.2828796750E-01
0.1431055982E+00 0.1003358539E+00
-0.6777711666E-01 -0.8695683128E-01
-0.1740980466E+00 -0.7322615732E-01
-0.9559132536E-01 0.9631505948E-01
0.1512981220E+00 0.2847632852E+00
0.6357421629E+00 -0.7284839004E+00
0.6823643465E+00 0.5708436334E+00
0.1963091141E+00 -0.5757775517E-02
-0.3301996982E-01 -0.1685142803E+00
0.2833562039E-01 0.2152759089E-01
0.2623688365E-01 0.1993310503E-01
0.7965029532E-02 0.1109055091E-01
-0.2796122002E-01 -0.3893335648E-01
0.1706238600E-01 0.2375776006E-01
0.2928655475E-01 0.4077875984E-01
0.2857806991E-02 0.3979226173E-02
-0.4686793277E-01 -0.6525916723E-01
-0.1702677471E+00 -0.3158187796E-01
0.3028147873E-01 -0.1797740855E+00
0.6351733467E+00 0.7214516731E+00
0.7207789094E+00 -0.6372838903E+00
0.1995469939E+00 0.1632013715E-01
-0.3864256419E-01 0.1755600827E+00
0.2947288315E-01 -0.2116688221E-01
0.2728991267E-01 -0.1959911299E-01
entries for the triangular preconditioning matrix and its inverse:
0.3743818971E+01 0.2671069322E+00
-0.6861305063E+01 0.1047409836E+01
0.7183286173E+01 -0.6066998614E+00
-0.3711751230E+01 0.2622361574E+00
0.7595392314E+00 -0.5336282441E-01
0.0000000000E+00 0.0000000000E+00
0.1749746932E+01 0.5715112179E+00
-0.1112386159E+01 0.5118280722E+00
0.5611031648E+00 -0.2021081048E+00
-0.1136294342E+00 0.3953339052E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1242099062E+01 0.8050887651E+00
-0.2151591879E+00 0.1662752175E+00
0.4303743724E-01 -0.3186979674E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1041780293E+01 0.9598952930E+00
-0.1641111826E-01 0.1572808312E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1001581378E+01 0.9984211186E+00
N=6
coefficients for the interior lowpass filter
(Beware: they are normalized so that their sum is 2)
- 0.0110318675094
0.00249992209279
0.06325056266
- 0.0297837512985
- 0.102724969862
0.477904371333
1.11389278393
0.694457972958
- 0.0683231215866
- 0.166863215412
0.00493661237185
0.0217847003266
coefficients for the edge filters:
0.9231184460E+00 -0.2321705918E-01
0.3781064509E+00 0.1171366871E+00
0.6815972396E-01 -0.3314072447E+00
-0.1063392935E-01 0.6083387113E+00
-0.9724819798E-02 -0.6313905276E+00
0.1297887318E-02 0.3213644020E+00
0.5723757426E-02 -0.6248310180E-01
-0.2868924125E+00 0.6092307458E-01
0.5940469387E+00 -0.2496185025E+00
0.6609604336E+00 0.4764189916E+00
0.3412988002E+00 -0.3565792957E+00
0.9716877076E-01 -0.2893047969E+00
-0.1860855577E-01 0.6336061692E+00
-0.4013700006E-01 -0.3069767199E+00
0.1537307481E-02 0.4772815467E-02
0.6783960390E-02 0.2106188350E-01
0.1472870932E+00 -0.1146309215E+00
-0.3468945123E+00 0.3715476805E+00
0.6221043773E-01 -0.4207564723E+00
0.4938022108E+00 -0.1034481165E+00
0.5849105895E+00 0.4067108104E+00
0.4598788833E+00 0.2977351919E+00
0.2274224529E+00 -0.6148599157E+00
-0.2788953680E-01 0.6853115007E-01
-0.6367943024E-01 0.1538100267E+00
0.2177711568E-02 -0.5448874481E-02
0.9609989671E-02 -0.2404525386E-01
-0.8324138021E-01 0.1150865862E+00
0.2035630921E+00 -0.2924528794E+00
-0.8289131955E-01 0.1456717540E+00
-0.1939904105E+00 0.2552553980E+00
-0.1908457968E-01 -0.1279750816E-02
0.2636403540E+00 -0.3051944288E+00
0.5123094150E+00 -0.4352292068E+00
0.6538041187E+00 0.7155631228E+00
0.3695601186E+00 -0.1145542755E+00
-0.3993953251E-01 -0.4300360725E-01
-0.9479455728E-01 -0.6987749084E-01
0.2986558645E-02 0.4395928052E-02
0.1317933842E-01 0.1939872286E-01
0.3164853364E-01 0.7165752442E-01
-0.8214471961E-01 -0.1869771835E+00
0.4327551059E-01 0.1021125523E+00
0.6830653932E-01 0.1533850430E+00
-0.4840048148E-02 -0.1330465569E-01
-0.7320834090E-01 -0.1616269775E+00
-0.5786918156E-01 -0.1222261416E+00
0.2055855208E-01 -0.2142989488E+00
0.4001944206E+00 0.8103733356E+00
0.7644983569E+00 -0.4058841524E+00
0.4653802512E+00 -0.1135045856E+00
-0.4686421163E-01 0.2621633969E-01
-0.1137136979E+00 0.6071714788E-01
0.3413297317E-02 -0.2015600401E-02
0.1506248690E-01 -0.8894611814E-02
-0.5550804971E-02 0.1092693197E-01
0.1471087926E-01 -0.2895882270E-01
-0.8859117912E-02 0.1743944876E-01
-0.1158923365E-01 0.2281376635E-01
0.1569832259E-02 -0.3090263591E-02
0.1157111171E-01 -0.2277809280E-01
0.7445526684E-02 -0.1465675051E-01
0.2902493903E-01 -0.8673422325E-01
-0.2999940975E-01 0.6576188609E-01
-0.6576340468E-01 0.4762105224E+00
0.3438258786E+00 -0.7968622388E+00
0.7867130346E+00 0.3399096026E+00
0.4895844920E+00 0.7597309556E-01
-0.4825468670E-01 -0.2118117647E-01
-0.1177993251E+00 -0.4513809001E-01
0.3488475665E-02 0.1772121221E-02
0.1539424027E-01 0.7820166308E-02
entries for the triangular preconditioning matrix and its inverse:
0.2691258238E+01 0.3715734097E+00
-0.4691390835E+01 0.7677024985E+00
0.5650878809E+01 -0.1862336274E+00
-0.4003610222E+01 0.4546833284E-01
0.1548107666E+01 -0.9293368857E-02
-0.2523403787E+00 0.1131936169E-02
0.0000000000E+00 0.0000000000E+00
0.2270666166E+01 0.4403993925E+00
-0.2346218035E+01 0.6446164864E+00
0.1686964492E+01 -0.2228021018E+00
-0.6557730032E+00 0.6332382245E-01
0.1069443363E+00 -0.8983535001E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1602926731E+01 0.6238588332E+00
-0.7485464376E+00 0.3995416825E+00
0.3030447729E+00 -0.1345491657E+00
-0.5083020320E-01 0.2195487513E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1168807480E+01 0.8555728955E+00
-0.1287884925E+00 0.1077442661E+00
0.2180567976E-01 -0.1782040660E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1022680347E+01 0.9778226434E+00
-0.7652542041E-02 0.7478034810E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1000641088E+01 0.9993593230E+00
N=7
coefficients for the interior lowpass filter:
(Beware: they are normalized so that their sum is 2)
0.014521394762
0.00567134268574
- 0.152463871896
- 0.198056706807
0.408183939725
1.08578270981
0.758162601964
0.0246656594886
- 0.070078291222
0.0960147679355
0.043155452582
- 0.0178704316511
- 0.0014812259146
0.0037926585342
coefficients for the edge filters:
0.9426568713E+00 0.2111640418E-01
0.2691475418E+00 -0.3198838656E+00
0.1685897688E+00 0.5837712653E+00
0.9379442214E-01 -0.5101698301E+00
0.4001600171E-01 0.4662343677E+00
0.1057520816E-01 -0.2666472827E+00
-0.1804517915E-02 0.8683768041E-01
0.4620417362E-02 -0.1225652919E-01
-0.3237338194E+00 -0.3337831390E-01
0.6746802222E+00 0.3515528841E+00
0.5985769136E+00 -0.4361153142E+00
0.2331427899E+00 0.2303567833E-01
0.7615188825E-02 0.3942229572E+00
0.8787238208E-01 -0.5996924995E+00
0.1327822780E+00 0.3966958788E+00
-0.4208296665E-01 -0.1101295389E+00
-0.4763766998E-02 -0.2001792798E-02
0.1219755973E-01 0.5125562862E-02
-0.2713311808E-01 -0.6003418428E-01
-0.2899110477E+00 0.4010347311E+00
0.6661404639E-01 -0.2255897141E+00
0.7251175151E+00 -0.3612133157E+00
0.5750729858E+00 0.4028777516E+00
0.4362186445E-01 0.2083956160E+00
-0.1857980605E+00 -0.6015619432E+00
0.1201045634E+00 0.2500040680E+00
0.5011061230E-01 0.1224078560E+00
-0.2991531277E-01 -0.9490678016E-01
-0.1573393605E-02 -0.3494310966E-02
0.4028652635E-02 0.8947135057E-02
0.2428297386E-01 -0.9808774833E-02
-0.1012534619E-01 0.2839107988E-01
-0.1589124403E+00 0.2026374721E-02
-0.1400872185E+00 0.3959072389E-02
0.3332116140E+00 -0.1570876545E-01
0.7053065273E+00 -0.9207675872E-01
0.5793528972E+00 -0.5918868498E-02
0.3896614469E-02 0.5375286822E+00
-0.5955471765E-01 -0.7665446854E+00
0.7479188863E-01 0.2868941871E+00
0.3081374376E-01 0.1404254891E+00
-0.1340206604E-01 -0.1078898335E+00
-0.1047350551E-02 -0.4024427396E-02
0.2681726647E-02 0.1030449087E-01
0.2396415492E-02 0.1254105463E-02
-0.2199441967E-01 -0.1041703156E-01
0.3831273498E-01 0.1308623534E-01
0.8416826933E-02 0.7187262673E-02
-0.1404553727E+00 -0.3166691268E-01
-0.9975512873E-01 -0.1089545516E-01
0.2671255526E+00 0.5917635073E-01
0.7725679637E+00 0.4986270055E-01
0.5400947777E+00 0.1873760767E-01
0.1395136806E-01 -0.5366643035E+00
-0.4947167749E-01 0.7678225002E+00
0.6809969285E-01 -0.2885552645E+00
0.3044230228E-01 -0.1401119926E+00
-0.1259630803E-01 0.1078385285E+00
-0.1045026849E-02 0.4012411756E-02
0.2675776840E-02 -0.1027372499E-01
-0.9493922980E-03 -0.4169893239E-04
0.8445740060E-02 0.4623987716E-03
-0.8462283108E-02 -0.6221335658E-03
-0.3100522289E-02 0.1572706697E-03
0.2269052507E-01 0.2991444009E-02
-0.9675176425E-02 0.4406571767E-03
-0.1013545517E+00 -0.1220793447E-01
-0.1405275956E+00 -0.3060471910E-01
0.2877816730E+00 0.6783517122E-01
0.7686169120E+00 0.4959733828E-01
0.5358841401E+00 0.1742614880E-01
0.1743190445E-01 -0.5361004996E+00
-0.4949055461E-01 0.7677690648E+00
0.6785067390E-01 -0.2886325963E+00
0.3050565786E-01 -0.1400474616E+00
-0.1262873733E-01 0.1078086397E+00
-0.1047102354E-02 0.4010247499E-02
0.2681091142E-02 -0.1026818344E-01
0.1364120169E-03 0.3562754358E-04
-0.1153014522E-02 -0.3011395265E-03
0.1050196147E-02 0.2742856482E-03
0.6396328807E-03 0.1670567176E-03
-0.1940525665E-02 -0.5068179562E-03
0.1962825019E-02 0.5126423314E-03
0.9431646581E-02 0.2463316874E-02
0.4027633417E-02 0.1051919469E-02
-0.1076854888E+00 -0.1260408191E-01
-0.1401281648E+00 -0.3053643220E-01
0.2886799721E+00 0.6790539526E-01
0.7677614604E+00 0.4955094521E-01
0.5360881498E+00 0.1743687297E-01
0.1744975917E-01 -0.5360995353E+00
-0.4954996630E-01 0.7677648691E+00
0.6789074612E-01 -0.2886301380E+00
0.3051526119E-01 -0.1400473018E+00
-0.1263602514E-01 0.1078082911E+00
-0.1047379021E-02 0.4010246532E-02
0.2681799545E-02 -0.1026818096E-01
entries for the triangular preconditioning matrix and its inverse:
0.5804269091E+00 0.1722869812E+01
-0.7323447716E+00 0.5738683253E+01
0.2824499599E+01 -0.1245382394E+02
-0.2917456460E+01 0.7964976943E+01
0.1789930270E+01 -0.4649550414E+01
-0.6097284388E+00 0.1520565899E+01
0.8862324360E-01 -0.2151255312E+00
0.0000000000E+00 0.0000000000E+00
0.2198648442E+00 0.4548248738E+01
0.5966667693E+00 -0.4076684813E+01
-0.9937306950E-01 0.1227270380E+01
0.5522044783E-01 -0.6678395557E+00
-0.1473788829E-01 0.1967618134E+00
0.1693029724E-02 -0.2546662797E-01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.6656852331E+00 0.1502211481E+01
0.1901881588E+00 -0.2856934692E+00
-0.1038800577E+00 0.1544684019E+00
0.3255433820E-01 -0.4820986746E-01
-0.4462431097E-02 0.6604466353E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1000032785E+01 0.9999672157E+00
-0.4315341245E-02 0.4305484704E-02
0.1542073127E-02 -0.1534917109E-02
-0.2143390832E-03 0.2132631237E-03
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1002256439E+01 0.9977486406E+00
-0.1554134553E-02 0.1550217349E-02
0.2204062611E-03 -0.2197888501E-03
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1000269825E+01 0.9997302475E+00
-0.7738658261E-04 0.7736527399E-04
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1000005602E+01 0.9999943980E+00
N=8
coefficients for the interior lowpass filter:
(beware: they are normalized so that their sum is 2)
0.00267279339281
- 0.000428394300246
- 0.0211456865284
0.00538638875377
0.0694904659113
- 0.0384935212634
- 0.0734625087609
0.515398670374
1.09910663054
0.68074534719
- 0.0866536154058
- 0.202648655286
0.0107586117505
0.0448236230437
- 0.000766690896228
- 0.0047834585115
coefficients for the edge filters:
0.9281136260E+00 -0.7274859222E-02
0.3668618286E+00 0.4042756257E-01
0.6266997447E-01 -0.1352706631E+00
-0.7647264163E-02 0.3196302046E+00
-0.5088004810E-02 -0.5501818613E+00
0.1171898539E-02 0.6205221664E+00
0.1656055473E-02 -0.4113728030E+00
-0.1929333000E-03 0.1430304841E+00
-0.1203653707E-02 -0.2044325554E-01
-0.2926634432E+00 0.2188690374E-01
0.6373078980E+00 -0.1065695633E+00
0.6452435440E+00 0.2843724182E+00
0.2965161471E+00 -0.4589840425E+00
0.5906254343E-01 0.3646732234E+00
-0.1366965641E-01 0.1447078292E+00
-0.1323318522E-01 -0.5672211027E+00
0.2213708145E-02 0.4500458402E+00
0.8306912600E-02 -0.1356216791E+00
-0.1892199826E-03 0.8474645355E-03
-0.1180561737E-02 0.5287413026E-02
0.1537394280E+00 0.4622773093E-01
-0.3933877381E+00 -0.1945832994E+00
0.1368645026E+00 0.3992968143E+00
0.6064470308E+00 -0.3762144385E+00
0.5685338138E+00 -0.7211460588E-01
0.3166230149E+00 0.4092764167E+00
0.9637901837E-01 0.5023569070E-01
-0.2215743488E-01 -0.5779911399E+00
-0.4105959919E-01 0.3834915553E+00
0.3062891878E-02 -0.1151747954E-01
0.1191510029E-01 -0.4999008246E-01
-0.2473506634E-03 0.7518452020E-03
-0.1543244666E-02 0.4690834793E-02
-0.9143117079E-01 -0.7950461567E-01
0.2571933236E+00 0.2823466127E+00
-0.1858137201E+00 -0.4114017970E+00
-0.2318000501E+00 0.1023384550E+00
0.1598859051E+00 0.3519831583E+00
0.4871009258E+00 -0.6162537699E-01
0.5734443542E+00 -0.4034315192E+00
0.4450618899E+00 -0.1544031905E+00
0.2033090498E+00 0.6034144392E+00
-0.3331081673E-01 -0.1069950598E+00
-0.6859620275E-01 -0.2019995950E+00
0.4406096842E-02 0.1468973030E-01
0.1763488599E-01 0.5758759998E-01
-0.3388224811E-03 -0.1171092606E-02
-0.2113946167E-02 -0.7306559816E-02
0.5709470789E-01 0.6367040385E-01
-0.1665850443E+00 -0.1917735089E+00
0.1454047216E+00 0.1859227361E+00
0.1015538648E+00 0.8760929009E-01
-0.1362852159E+00 -0.1698024241E+00
-0.1769345243E+00 -0.1650717698E+00
0.8596479652E-02 0.4044496899E-01
0.3062987789E+00 0.2651503207E+00
0.5431752460E+00 0.3382384364E+00
0.6140955155E+00 -0.7711604374E+00
0.3284966144E+00 0.2754111726E+00
-0.4723534035E-01 0.4710140765E-01
-0.1039762360E+00 0.2958356475E-01
0.6053819361E-02 -0.8186932177E-02
0.2473853830E-01 -0.2780308340E-01
-0.4480360998E-03 0.8002315701E-03
-0.2795340476E-02 0.4992722014E-02
-0.2771885178E-01 0.5905246892E-01
0.8389262409E-01 -0.1800708576E+00
-0.8147463473E-01 0.1795432638E+00
-0.3866640174E-01 0.7542773595E-01
0.7464550842E-01 -0.1632828547E+00
0.6920099668E-01 -0.1404076028E+00
-0.2231653624E-01 0.5304071283E-01
-0.1020129060E+00 0.2049015597E+00
-0.6275852804E-01 0.1129722066E+00
0.8683887629E-01 0.6297984646E-01
0.4540741128E+00 -0.7409643915E+00
0.7328223558E+00 0.5042713730E+00
0.4319437621E+00 0.7588728111E-01
-0.5734179466E-01 -0.4120932567E-01
-0.1314896433E+00 -0.8313424682E-01
0.7195965493E-02 0.5540498594E-02
0.2979946085E-01 0.2315912504E-01
-0.5190323961E-03 -0.3922287088E-03
-0.3238293222E-02 -0.2447152777E-02
0.8529185218E-02 0.1789547417E-01
-0.2624362023E-01 -0.5515123460E-01
0.2697489980E-01 0.5699017157E-01
0.9748173003E-02 0.1992423503E-01
-0.2431726904E-01 -0.5131208900E-01
-0.1901319060E-01 -0.3947317762E-01
0.8695522469E-02 0.1852759787E-01
0.2729552258E-01 0.5658785929E-01
0.1231155208E-01 0.2487421369E-01
0.9187597846E-02 0.5732429572E-01
-0.4745631970E-01 -0.1020173091E+00
-0.2924026466E-01 -0.4251128534E+00
0.3831732042E+00 0.8001954561E+00
0.7723764353E+00 -0.3763869007E+00
0.4743837775E+00 -0.6582574501E-01
-0.6082976218E-01 0.2842730285E-01
-0.1418145038E+00 0.5331847291E-01
0.7565491348E-02 -0.3918460367E-02
0.3149214931E-01 -0.1548884491E-01
-0.5401012058E-03 0.3080051475E-03
-0.3369743560E-02 0.1921673848E-02
-0.1148179452E-02 0.2054346689E-02
0.3567347972E-02 -0.6382774933E-02
-0.3784785333E-02 0.6771819551E-02
-0.1105680377E-02 0.1978304192E-02
0.3387180654E-02 -0.6060419089E-02
0.2395725348E-02 -0.4286482122E-02
-0.1281174092E-02 0.2292309675E-02
-0.3407989030E-02 0.6097652254E-02
-0.1283416571E-02 0.2296316381E-02
-0.9841020551E-02 0.2254192382E-01
0.6341196709E-02 -0.1213654065E-01
0.4619539219E-01 -0.1379967819E+00
-0.2943416082E-01 0.6522286075E-01
-0.5094613096E-01 0.4794644197E+00
0.3655480429E+00 -0.7789628537E+00
0.7770499808E+00 0.3646968778E+00
0.4810695095E+00 0.5250853492E-01
-0.6126039684E-01 -0.2724349082E-01
-0.1432436897E+00 -0.4923361338E-01
0.7606483137E-02 0.3810616823E-02
0.3168981524E-01 0.1496216731E-01
-0.5420982042E-03 -0.3029798461E-03
-0.3382203026E-02 -0.1890320507E-02
entries for the triangular preconditioning matrix and its inverse:
0.2810123818E+01 0.3558562059E+00
-0.6559223197E+01 0.8146829794E+00
0.1166265739E+02 -0.1366545660E+00
-0.1370803617E+02 -0.2512978838E-01
0.1057161896E+02 0.3435512306E-01
-0.5161014832E+01 -0.1989050428E-01
0.1450032681E+01 0.6413814665E-02
-0.1790147104E+00 -0.8795283072E-03
0.0000000000E+00 0.0000000000E+00
0.2865090274E+01 0.3490291420E+00
-0.4726642907E+01 0.7527005851E+00
0.5658991531E+01 -0.1741151174E+00
-0.4391120800E+01 0.2189754588E-01
0.2150489727E+01 0.7628539402E-02
-0.6053327281E+00 -0.4888651292E-02
0.7483245389E-01 0.8230905056E-03
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.2191756126E+01 0.4562551409E+00
-0.2253964453E+01 0.6427198947E+00
0.1808275069E+01 -0.2393601344E+00
-0.9018774144E+00 0.8134531543E-01
0.2565106244E+00 -0.1922603801E-01
-0.3190347407E-01 0.2116725807E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1600048291E+01 0.6249811369E+00
-0.8330291569E+00 0.4302638030E+00
0.4321721196E+00 -0.1746850867E+00
-0.1262794856E+00 0.4820621391E-01
0.1601926411E-01 -0.6049455037E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1210019309E+01 0.8264330929E+00
-0.2036886238E+00 0.1611623644E+00
0.6037258981E-01 -0.4551697161E-01
-0.7732501809E-02 0.5803277507E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1044505769E+01 0.9573905957E+00
-0.2609642521E-01 0.2489086650E-01
0.3366051566E-02 -0.3198214876E-02
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1003760640E+01 0.9962534497E+00
-0.9726417449E-03 0.9689366947E-03
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1000062955E+01 0.9999370493E+00
*****************************************************************
TABLES FOR THE RIGHT EDGE.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
The coefficients for the interior filters are exactly the same as
above for every value of N, so that we do not repeat them here.
N=2
coefficients for the edge filters:
0.8705087534E+00 -0.2575129195E+00
0.4348969980E+00 0.8014229620E+00
0.2303890438E+00 -0.5398225007E+00
-0.1942334074E+00 0.3717189665E+00
0.1901514184E+00 -0.3639069596E+00
0.3749553316E+00 -0.7175799994E+00
0.7675566693E+00 0.4010695194E+00
0.4431490496E+00 0.2315575950E+00
entries for the triangular preconditioning matrix and its inverse:
0.1089843053E+01 0.9175633109E+00
0.0000000000E+00 0.0000000000E+00
-0.8008132342E+00 0.3505220326E+00
0.2096292884E+01 0.4770325785E+00
N=3
coefficients for the edge filters:
0.9096849943E+00 0.7221948764E-01
0.3823606559E+00 -0.4265622188E+00
0.1509872153E+00 0.8042331410E+00
0.5896101069E-01 -0.4074776976E+00
-0.2904078511E+00 -0.1535052307E+00
0.4189992290E+00 0.5223942379E+00
0.4969643721E+00 -0.9819800088E-01
0.4907578307E+00 -0.7678795743E+00
0.4643627674E+00 0.2985152397E+00
0.1914505442E+00 0.1230738317E+00
0.8183541840E-01 0.2294775484E+00
-0.1587582156E+00 -0.4451794444E+00
-0.9124735623E-01 -0.2558698912E+00
0.6042558204E-03 0.1694414802E-02
0.7702933610E-01 0.7598761510E+00
0.5200601778E+00 0.1391503267E+00
0.7642591993E+00 -0.2725472352E+00
0.3150938230E+00 -0.1123675716E+00
entries of the triangular preconditioning matrix and its inverse:
0.1055782528E+01 0.9471647557E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.4658798302E+00 0.2539461793E+00
0.1737631796E+01 0.5754959149E+00
0.0000000000E+00 0.0000000000E+00
0.2417783151E+01 -0.1059694498E+00
-0.6663367942E+01 0.6796520006E+00
0.5642212524E+01 0.1772354366E+00
N=4
coefficients for the edge filters:
0.9154705188E+00 -0.1982779906E+00
0.3919142810E+00 0.6040677868E+00
0.5947771124E-01 -0.6495297603E+00
-0.2519180851E-01 0.4050309541E+00
0.6437934569E-01 -0.9924194741E-01
-0.2191626469E+00 -0.2726273506E+00
0.4488001781E+00 0.5092867484E+00
0.7540005084E+00 0.6811856626E-01
0.3937758157E+00 -0.6735345752E+00
-0.1581338944E+00 0.4499340827E+00
-0.1614201190E-01 0.2719065540E-01
0.4126840881E-01 -0.6951519362E-01
0.1290078289E-01 0.4581959340E-02
-0.1390716006E+00 0.3062032365E-01
0.2921367950E-01 0.1388737145E-01
0.4606168537E+00 -0.9504834999E-01
0.8164119742E+00 -0.3015815012E+00
0.2986473346E+00 0.8033636317E+00
-0.1027663536E+00 -0.4968427065E+00
-0.1257488211E-01 -0.2963204370E-01
0.3214874197E-01 0.7575680778E-01
-0.6775603652E-02 0.2880305124E-02
0.1913244129E-01 -0.8133189531E-02
-0.1770918425E-01 0.7528163799E-02
-0.6765916174E-01 0.2876186983E-01
-0.3023588481E-01 0.1285325684E-01
0.4977920821E+00 -0.9920191439E-01
0.8039495996E+00 -0.2977899841E+00
0.2977111011E+00 0.8037936841E+00
-0.9910804055E-01 -0.4976470524E+00
-0.1259895190E-01 -0.2963766018E-01
0.3221027840E-01 0.7577116678E-01
entries of the triangular preconditioning matrix and its inverse:
0.1000398078E+01 0.9996020804E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.2241154296E-02 0.2235092825E-02
0.1002312956E+01 0.9976923812E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.1844504727E-01 0.2335082980E-01
0.9170462790E-01 -0.1171759021E+00
0.7808176166E+00 0.1280708809E+01
0.0000000000E+00 0.0000000000E+00
-0.7373304912E-02 -0.2846460011E-02
-0.9310068502E-03 0.9005357890E-01
0.3767386370E+00 -0.9639839213E+00
0.5005192311E+00 0.1997925230E+01
N=5
coefficients of the edge filters:
0.6629994791E+00 0.2704377396E+00
0.5590853114E+00 -0.2099389697E+00
-0.3638895712E+00 0.6232723378E+00
-0.3190292420E+00 -0.6204577645E+00
-0.8577516141E-01 0.3239177989E+00
0.7938922949E-01 -0.6655879286E-01
-0.1615976602E+00 -0.4460509080E+00
0.7429922432E+00 0.2531551137E+00
0.5771521705E+00 -0.3811235714E+00
0.2773360273E+00 -0.2939476383E+00
0.1530007177E-01 0.6480763210E+00
-0.1063942878E+00 -0.2856775435E+00
-0.1216769629E-01 -0.4448843004E-01
0.1126647052E-01 0.4119330180E-01
0.2576681838E+00 -0.4091805529E+00
-0.3995181391E-01 0.1427046841E+00
0.1290658921E-01 0.2088087907E-01
0.4163708782E+00 -0.3899540483E+00
0.6469374504E+00 -0.2052536834E+00
0.5608421202E+00 0.7190940497E+00
0.4995700940E-02 -0.3116714966E+00
-0.1569482750E+00 -0.1763286609E-01
-0.2009485730E-01 -0.4039743821E-01
0.1860648984E-01 0.3740531781E-01
-0.8955158953E-01 -0.1414833249E+00
0.3895616532E-01 0.6067058589E-01
0.4751157158E-02 -0.1094361815E-01
-0.6596188737E-01 -0.1401642518E+00
0.1520881337E-01 -0.7844525367E-01
0.2466644601E+00 0.5612750679E-01
0.7199998123E+00 0.5913073605E+00
0.6131240289E+00 -0.7443064557E+00
0.1537319456E-01 0.1990571237E+00
-0.1721947543E+00 0.4730213481E-01
-0.2083858871E-01 0.3055544136E-01
0.1929513523E-01 -0.2829228895E-01
0.1546583496E-01 0.1104876084E-01
-0.6559247482E-02 -0.4685912973E-02
0.2728093541E-02 0.1948944437E-02
0.1830645747E-01 0.1307809574E-01
0.1708296448E-01 0.1220403484E-01
0.2156026058E-01 0.1540260595E-01
-0.3692524465E-01 -0.1737014260E+00
0.2045334663E+00 -0.1298838460E-01
0.7233635779E+00 0.6336000391E+00
0.6327590443E+00 -0.7243137518E+00
0.1649999072E-01 0.1992776018E+00
-0.1751674471E+00 0.3929252813E-01
-0.2109311508E-01 0.2952572161E-01
0.1953080958E-01 -0.2733883754E-01
entries of the triangular preconditioning matrix and its inverse:
0.1000413355E+01 0.9995868157E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.4572991430E-02 0.4514077458E-02
0.1012632589E+01 0.9875250029E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.9186034191E-02 -0.8522720044E-02
-0.5149118250E-01 0.4842233071E-01
0.1050111166E+01 0.9522801325E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.8653411835E-01 0.4412830877E-01
0.4462922645E+00 -0.2280710782E+00
-0.9332961770E+00 0.5124746278E+00
0.1734250554E+01 0.5766179504E+00
0.0000000000E+00 0.0000000000E+00
-0.6651201997E-01 0.2197171709E+00
0.3272287057E+00 -0.1071279051E+01
-0.6835308959E+00 0.2403809220E+01
0.1157116986E+00 -0.2710986078E+00
0.2461150317E+00 0.4063140690E+01
N=6
coefficients of the edge filters:
0.9236675275E+00 -0.5565971353E-01
0.3793343885E+00 0.2242498085E+00
0.4986459778E-01 -0.5135424238E+00
-0.1688947163E-01 0.6321916480E+00
-0.3311806998E-02 -0.4901838615E+00
0.2804629640E-02 0.2042322283E+00
-0.1237553624E-01 -0.3508415169E-01
-0.2831955072E+00 0.1057159278E+00
0.6209232596E+00 -0.3394364574E+00
0.6494435367E+00 0.4787854504E+00
0.3279016732E+00 -0.7187645472E-01
0.4613465423E-01 -0.5042780615E+00
-0.2201105042E-01 0.5796966704E+00
0.4772996268E-01 -0.2205355783E+00
0.1811083865E-02 -0.3313371482E-02
-0.7992103956E-02 0.1462152573E-01
0.1415889950E+00 0.1743290418E+00
-0.3264117354E+00 -0.4255531446E+00
0.5898718517E-01 0.2638047100E+00
0.5662575494E+00 0.3616438204E+00
0.6511436245E+00 -0.2644671055E+00
0.3434709841E+00 -0.4908547063E+00
-0.6897565912E-01 0.4962107578E+00
-0.2139852264E-01 0.6828427694E-01
0.4514540543E-01 -0.1606899475E+00
0.1807019829E-02 -0.5156673789E-02
-0.7974169836E-02 0.2275580595E-01
-0.4080929146E-01 0.3244709516E-02
0.1251977353E+00 0.5111253513E-03
-0.9208945297E-01 0.4381492218E-02
-0.1815441604E+00 -0.3035847066E-01
0.7492893632E-01 -0.3796933443E-01
0.4921890166E+00 0.7322186910E-01
0.7598885402E+00 0.3436818708E+00
0.3298535864E+00 -0.7867428461E+00
-0.6885203537E-01 0.4886190401E+00
-0.2056224689E-01 0.4823811158E-01
0.4357038095E-01 -0.1176377166E+00
0.1729458762E-02 -0.3491719268E-02
-0.7631901801E-02 0.1540855391E-01
0.1473338385E-01 -0.5211034254E-02
-0.3879928306E-01 0.1379054345E-01
0.3222685441E-01 -0.1172287487E-01
0.4181351746E-01 -0.1651839644E-01
-0.4470160402E-01 0.1484603300E-01
-0.1143592801E+00 0.4328769202E-01
-0.3079264597E-01 0.1491923547E-01
0.4953303341E+00 -0.7352728683E-01
0.7831063037E+00 -0.3353441076E+00
0.3365261769E+00 0.7880177709E+00
-0.7161554542E-01 -0.4916362852E+00
-0.2096645980E-01 -0.4832809329E-01
0.4445995152E-01 0.1180259590E+00
0.1762244193E-02 0.3492043263E-02
-0.7776580122E-02 -0.1540998367E-01
-0.2279778462E-02 0.1154376053E-02
0.5934267127E-02 -0.3004842779E-02
-0.4653920620E-02 0.2356533579E-02
-0.4697326198E-02 0.2378512204E-02
0.7905498252E-02 -0.4002984474E-02
0.1478426761E-01 -0.7486080223E-02
0.6356609850E-03 -0.3218698441E-03
-0.1184137622E+00 0.4493383491E-01
-0.4719524338E-01 0.2049260653E-01
0.4914056494E+00 -0.7279363945E-01
0.7873827096E+00 -0.3377604925E+00
0.3378777169E+00 0.7876649023E+00
-0.7256793624E-01 -0.4910841873E+00
-0.2105690525E-01 -0.4831332253E-01
0.4471373463E-01 0.1179951690E+00
0.1767573251E-02 0.3490782271E-02
-0.7800096641E-02 -0.1540441905E-01
entries of the triangular preconditioning matrix and its inverse:
0.1000078420E+01 0.9999215862E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.9850976530E-03 0.9819736622E-03
0.1003102675E+01 0.9969069215E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.2633468462E-02 -0.2560290665E-02
-0.1664138984E-01 0.1623091263E-01
0.1022118540E+01 0.9783601031E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.1429038400E-02 0.1388671224E-02
0.9368484963E-02 -0.9150173851E-02
-0.2392821028E-01 0.2393097496E-01
0.9782470759E+00 0.1022236636E+01
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1195277199E-01 -0.1168953970E-01
-0.6711312742E-01 0.6542835502E-01
0.1479375368E+00 -0.1465646909E+00
-0.7026196422E-01 0.7358661282E-01
0.9760519092E+00 0.1024535673E+01
0.0000000000E+00 0.0000000000E+00
-0.1248032832E-01 0.2141640226E-01
0.7502310126E-01 -0.1275021840E+00
-0.1804736950E+00 0.3095551764E+00
0.1461500318E+00 -0.2545824950E+00
0.1504596966E+00 -0.2445553979E+00
0.6303329548E+00 0.1586463142E+01
N=7
coefficients for the edge filters:
0.9091449027E+00 0.1058599696E-01
0.4122956702E+00 -0.4261024318E-01
0.8463996964E-02 0.1626466305E+00
-0.4985469571E-01 -0.3983434261E+00
-0.2885243693E-01 0.6359366721E+00
-0.8468475604E-02 -0.5811383289E+00
0.9290995299E-03 0.2621339682E+00
0.2377975298E-02 -0.4571068446E-01
-0.2699308042E+00 -0.3369311938E-01
0.6287804833E+00 0.1173158297E+00
0.6425124921E+00 -0.3345684650E+00
0.3295936378E+00 0.4993594177E+00
0.9680040842E-01 -0.2270999192E+00
-0.1505090528E-02 -0.4092454872E+00
-0.2576996713E-01 0.5883989035E+00
-0.1663793375E-01 -0.2415648560E+00
0.7886583178E-03 0.4918478872E-02
0.2019348750E-02 0.1259369707E-01
0.2157402688E+00 -0.6868038150E-01
-0.3660961628E+00 0.2059784468E+00
0.9979560410E-01 -0.4260199250E+00
0.5343306371E+00 0.3030769028E+00
0.5651956498E+00 0.2528028219E+00
0.3984651316E+00 -0.2995453713E+00
0.2006608171E+00 -0.4153788373E+00
0.3790996947E-01 0.5783273640E+00
-0.5279896883E-01 -0.1200871578E+00
-0.3509616877E-01 -0.6583871196E-01
0.1600620645E-02 0.3864118919E-02
0.4098367097E-02 0.9894023233E-02
-0.1219999250E+00 0.1199301213E+00
0.2497494925E+00 -0.3021493088E+00
-0.1489636838E+00 0.4092549285E+00
-0.2314941689E+00 0.2642183651E-01
0.4306729538E-01 -0.3532455770E+00
0.3349988297E+00 -0.2065478346E+00
0.5170449138E+00 0.2227171845E+00
0.5314908191E+00 0.5240786066E+00
0.3984867666E+00 -0.2234158448E+00
0.1134176909E+00 -0.3876543947E+00
-0.8664578083E-01 0.1720771534E+00
-0.6206801690E-01 0.6600629896E-01
0.2555164772E-02 -0.5990160538E-02
0.6542464174E-02 -0.1533772348E-01
0.6643773700E-01 0.4081778499E-01
-0.1425076588E+00 -0.7665915987E-01
0.1150002072E+00 0.2089854289E-01
0.1110261320E+00 0.1140640809E+00
-0.5456405522E-01 0.1937409327E-01
-0.1403852745E+00 -0.1187989690E+00
-0.9002913124E-01 -0.1838791619E+00
0.9927107272E-01 -0.1133363121E+00
0.3252695570E+00 0.7805202816E+00
0.5955739150E+00 -0.5430653297E+00
0.6177120462E+00 0.6990288263E-01
0.2102093627E+00 -0.6927193818E-01
-0.1207077310E+00 0.6202160209E-01
-0.9029360085E-01 0.1264388999E-01
0.3498469159E-02 -0.2337275009E-02
0.8957782052E-02 -0.5984560439E-02
-0.2472698999E-01 0.1089534551E+00
0.5451727819E-01 -0.2418936349E+00
-0.4913268141E-01 0.2243741754E+00
-0.3543504434E-01 0.1488992192E+00
0.2981657384E-01 -0.1382661605E+00
0.5443081267E-01 -0.2312770306E+00
0.3080352023E-01 -0.1080114084E+00
-0.2311127518E-01 0.1488311123E+00
-0.2663146891E-01 0.2352279421E+00
-0.3986350788E-01 0.4020354092E+00
0.1012109980E+00 -0.7029809958E+00
0.5654368499E+00 -0.6175503239E-02
0.7419995864E+00 0.1485821868E+00
0.2730680885E+00 0.1174257523E+00
-0.1373359727E+00 -0.5325615968E-01
-0.1050857718E+00 -0.2772249506E-01
0.3942767529E-02 0.1737252935E-02
0.1009540190E-01 0.4448212190E-02
0.4060965940E-02 0.1545016950E-01
-0.9082349871E-02 -0.3455430249E-01
0.8675066430E-02 0.3300477019E-01
0.5263347825E-02 0.2002469773E-01
-0.5426164014E-02 -0.2064414148E-01
-0.8281419894E-02 -0.3150712024E-01
-0.2928559727E-02 -0.1114186730E-01
0.7407888982E-02 0.2818372382E-01
0.1392100926E-02 -0.5377738331E-01
0.2865053765E-01 0.1320738411E+00
0.5527738563E-01 0.2388167237E+00
-0.5250033174E-01 -0.7743478585E+00
0.2337385282E-01 0.5554662583E+00
0.5390029057E+00 -0.7133893666E-02
0.7666684946E+00 -0.5462109919E-01
0.2878132266E+00 -0.7104460368E-01
-0.1399692009E+00 0.3084906050E-01
-0.1077108034E+00 0.1309957900E-01
0.4008607630E-02 -0.1053633670E-02
0.1026398458E-01 -0.2697814488E-02
entries of the triangular preconditioning matrix and its inverse:
0.1000408431E+01 0.9995917354E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.5293682422E-02 0.5202484794E-02
0.1017114208E+01 0.9831737600E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1591493727E-01 -0.1341009335E-01
-0.1031512206E+00 0.8847321480E-01
0.1146285615E+01 0.8723829270E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.3646661500E-01 0.1680436654E-01
0.2406286152E+00 -0.1136375469E+00
-0.6581590373E+00 0.3658356512E+00
0.1569466249E+01 0.6371592893E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.8620589496E-01 -0.7264200771E-02
-0.5839504132E+00 0.5536106388E-01
0.1663842388E+01 -0.2090053065E+00
-0.2536273536E+01 0.6450028470E+00
0.2505431179E+01 0.3991328951E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.3341753604E+00 -0.3790478410E-02
0.2298323873E+01 0.1873956341E-01
-0.6712323867E+01 -0.2295905912E-01
0.1070192046E+02 -0.9239613787E-01
-0.9843372614E+01 0.7726440127E+00
0.5084895171E+01 0.1966608881E+00
0.0000000000E+00 0.0000000000E+00
0.1653022182E+00 0.3026314320E-01
-0.1367863363E+01 -0.1720350030E+00
0.4921932660E+01 0.4114532707E+00
-0.1002831667E+02 -0.6057060145E+00
0.1256338987E+02 0.5848923814E+00
-0.9758759367E+01 0.4444537862E+00
0.4318033378E+01 0.2315869083E+00
N=8
coefficients for the edge filters:
0.9280961080E+00 -0.1623137602E-01
0.3685482772E+00 0.7700181909E-01
0.5169019500E-01 -0.2216538064E+00
-0.1072856966E-01 0.4518914876E+00
-0.3547264354E-02 -0.5968036841E+00
0.1881724221E-02 0.5361053678E+00
0.2253426546E-03 -0.2972611580E+00
-0.4074549423E-03 0.9152878068E-01
0.2542121693E-02 -0.1206728019E-01
-0.2880715337E+00 0.3847998474E-01
0.6422316271E+00 -0.1590369267E+00
0.6540393284E+00 0.3562112107E+00
0.2730170577E+00 -0.4595861878E+00
0.4300012686E-01 0.1637283171E+00
-0.1532507378E-01 0.3534175472E+00
-0.3095082211E-02 -0.5814453781E+00
0.3081063250E-02 0.3686477291E+00
-0.1192117285E-01 -0.9703824753E-01
-0.2510321213E-03 -0.5800646631E-03
0.1566213637E-02 0.3619079426E-02
0.1554699662E+00 -0.7121179707E-01
-0.3928508188E+00 0.2523023478E+00
0.1655006354E+00 -0.4171714520E+00
0.6186900270E+00 0.2451267006E+00
0.5757733437E+00 0.2725111911E+00
0.2765573214E+00 -0.3518314346E+00
0.4215137090E-01 -0.2344734683E+00
-0.2022141803E-01 0.5903810045E+00
0.3170527734E-01 -0.3062645760E+00
0.2969565406E-02 -0.9326190637E-02
-0.1133537854E-01 0.4097674398E-01
-0.2472629557E-03 0.5916913076E-03
0.1542697449E-02 -0.3691619185E-02
-0.9273812077E-01 0.1179909741E+00
0.2525480154E+00 -0.3454318691E+00
-0.1903081753E+00 0.3669763444E+00
-0.2325263183E+00 0.8571956937E-01
0.1951123754E+00 -0.3733637204E+00
0.5747752998E+00 -0.1532441521E+00
0.6031889613E+00 0.3576146478E+00
0.3197987067E+00 0.3379298220E+00
-0.2318110629E-01 -0.4826776512E+00
-0.2335276594E-01 -0.1247229279E+00
0.3904642048E-01 0.2577717071E+00
0.3358324305E-02 0.1649336963E-01
-0.1296942483E-01 -0.6672983676E-01
-0.2744734004E-03 -0.1243663709E-02
0.1712466040E-02 0.7759337931E-02
0.4096472123E-01 0.9544232866E-02
-0.1240767851E+00 -0.2688549916E-01
0.1278114403E+00 0.3303656538E-01
0.7138489316E-01 -0.1099762148E-01
-0.1584871307E+00 -0.3331736763E-01
-0.1727866355E+00 0.3186523735E-01
0.1143640280E+00 0.7533252701E-01
0.5129116028E+00 -0.5202591621E-01
0.7137757767E+00 -0.3899644872E+00
0.3414317954E+00 0.7710342652E+00
-0.3881329228E-01 -0.4656907083E+00
-0.2526829049E-01 -0.6031146457E-01
0.4424255303E-01 0.1391049995E+00
0.3575822034E-02 0.7543265904E-02
-0.1394488775E-01 -0.3124469163E-01
-0.2875903358E-03 -0.5438414561E-03
0.1794303867E-02 0.3393079342E-02
-0.1790191282E-01 -0.5044968595E-02
0.5295161530E-01 0.1488052556E-01
-0.5704547538E-01 -0.1581581210E-01
-0.1894676577E-01 -0.6949272400E-02
0.6968256270E-01 0.2057861785E-01
0.4547075753E-01 0.1765306809E-01
-0.7337937479E-01 -0.1862118927E-01
-0.1444748191E+00 -0.4891212508E-01
-0.1409191494E-01 -0.1375364073E-01
0.4948704363E+00 0.5400109528E-01
0.7618895951E+00 0.3573536308E+00
0.3591100053E+00 -0.7782672053E+00
-0.4810103061E-01 0.4837479787E+00
-0.2674149513E-01 0.6134832074E-01
0.4783092078E-01 -0.1436058269E+00
0.3754819526E-02 -0.7612327455E-02
-0.1470868404E-01 0.3171165160E-01
-0.2997260590E-03 0.5426010792E-03
0.1870019815E-02 -0.3385340511E-02
0.4842573875E-02 0.2223331313E-02
-0.1435818114E-01 -0.6588142232E-02
0.1564773617E-01 0.7160556501E-02
0.3635704163E-02 0.1734300817E-02
-0.1807616145E-01 -0.8390109930E-02
-0.7951972463E-02 -0.3916508325E-02
0.2177252927E-01 0.9923280681E-02
0.3159516732E-01 0.1489081491E-01
-0.5393073080E-02 -0.2164190023E-02
-0.1454356948E+00 -0.5005924913E-01
-0.5484912621E-01 -0.2423995567E-01
0.4834662634E+00 0.5271872910E-01
0.7753953496E+00 0.3632737173E+00
0.3639514649E+00 -0.7773694792E+00
-0.5141318290E-01 0.4816288882E+00
-0.2717238099E-01 0.6128840935E-01
0.4899043649E-01 -0.1433368916E+00
0.3804055936E-02 -0.7609148826E-02
-0.1492982098E-01 0.3170285605E-01
-0.3026846009E-03 0.5422216523E-03
0.1888478444E-02 -0.3382973230E-02
-0.5886912723E-03 0.3289309542E-03
0.1750384584E-02 -0.9780261437E-03
-0.1931250902E-02 0.1079085355E-02
-0.3621632497E-03 0.2023585594E-03
0.2085884179E-02 -0.1165486300E-02
0.6409242844E-03 -0.3581158117E-03
-0.2736289413E-02 0.1528899888E-02
-0.3368917714E-02 0.1882379797E-02
0.1038171472E-02 -0.5800785496E-03
0.3193936333E-01 -0.1508848895E-01
0.6823471193E-02 -0.3370611049E-02
-0.1435014917E+00 0.4925184726E-01
-0.6098305316E-01 0.2705640950E-01
0.4814477634E+00 -0.5199195925E-01
0.7771077176E+00 -0.3643916717E+00
0.3644295324E+00 0.7771916813E+00
-0.5192289139E-01 -0.4813707824E+00
-0.2721779258E-01 -0.6127396434E-01
0.4913249792E-01 0.1432965635E+00
0.3808651449E-02 0.7607535473E-02
-0.1495173150E-01 -0.3169534309E-01
-0.3029170562E-03 -0.5421338829E-03
0.1889928755E-02 0.3382425628E-02
entries of the triangular preconditioning matrix and its inverse:
0.1000011417E+01 0.9999885829E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.1885821248E-03 0.1884331061E-03
0.1000779405E+01 0.9992212023E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.6980138374E-03 -0.6895681425E-03
-0.5775869059E-02 0.5710508693E-02
0.1010657918E+01 0.9894544755E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.1587420489E-02 0.1473040141E-02
0.1315517251E-01 -0.1221739122E-01
-0.4838107469E-01 0.4544822540E-01
0.1053305612E+01 0.9493920742E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.2302522936E-02 -0.1832761923E-02
-0.1919511889E-01 0.1530411869E-01
0.7113262876E-01 -0.5742370539E-01
-0.1541810491E+00 0.1326319561E+00
0.1103642518E+01 0.9060904992E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
-0.6297296878E-02 0.3996745292E-02
0.5008238228E-01 -0.3139508197E-01
-0.1744341513E+00 0.1085741195E+00
0.3478392588E+00 -0.2247406296E+00
-0.4139824420E+00 0.3061849704E+00
0.1225094612E+01 0.8162634872E+00
0.0000000000E+00 0.0000000000E+00
0.0000000000E+00 0.0000000000E+00
0.1205751485E-01 -0.5357488581E-02
-0.9775783312E-01 0.4355462708E-01
0.3469638042E+00 -0.1557775571E+00
-0.7026319735E+00 0.3312968007E+00
0.8627444834E+00 -0.4752264599E+00
-0.6802055242E+00 0.4601200251E+00
0.1206700215E+01 0.8287062417E+00
0.0000000000E+00 0.0000000000E+00
-0.1638857751E-01 0.1062802918E-01
0.1318974327E+00 -0.8500762499E-01
-0.4651165383E+00 0.2996885100E+00
0.9374435384E+00 -0.6305890566E+00
-0.1161167601E+01 0.9184100559E+00
0.8600562361E+00 -0.8062512474E+00
-0.2230327919E+00 0.2486081011E+00
0.7434539179E+00 0.1345073280E+01
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
***************************************************************************
TABLES FOR WAVELETS ON THE INTERVAL IN THE CONSTRUCTION
OF YVES MEYER (COMPUTED BY PIERRE VIAL)
***************************************************************************
These tables correspond to a different construction for wavelets on
the interval, due to Yves Meyer (Revista Matematica Iberoamericana,
7 (1992) p. 115-133). The details of how these tables were obtained,
as well as graphs corresponding to the cases N=2 and N=4, can be found
in "Wavelets on the interval and fast algorithms", by Albert Cohen,
Ingrid Daubechies and Pierre Vial, to appear in Journal for Applied
and Computational Harmonic Analysis (1993).
The organization of these tables is as follows:
- table of the interior filter coefficients for the different values
of N. We have again chosen the filters with closest to linear
phase, leading to "near-symmetric" scaling functions and wavelets.
Note: the interior coefficients given are not always normalized so that
their sum is sqrt(2). They should be renormalized before
they are used.
These interior filters are the same as in the first set of
tables given above, but for some values of N their order has
been inverted. This will mean that the graphs are mirror images.
If you would like the same graphs (without mirroring), then
you should invert the order of the interior coefficients,
exchange the right edge table with the left edge table (for
this value of N), and invert the order of all the edge coefficients
in each subgroup as well.
- table of edge coefficients for the scaling functions ("low pass" filters)
at the left side, for N=2 to 8
- table of edge coefficients for the wavelet ("high pass" filters) at the
left side, for N=2 to 8
- table of edge coefficients for the scaling functions ("low pass" filters)
at the right side, for N=2 to 8
- table of edge coefficients for the wavelet ("high pass" filters) at the
right side, for N=2 to 8
Note: the tables of the right side have been obtained by doing exactly
the same computations, and listing the results in the same order, as for
the left side tables, but with the order of the interior coefficients
inverted.
**************************************************************************
The interior coefficients in these tables were computed with Macsyma.
The edge coefficients were then computed, using the interior filter
coefficients, by means of a Fortran program using double precision
for N=2,3,4,5 and quadruple precision for N=6,7,8 .
***************************************************************************
INTERIOR COEFFICIENTS
*********************
N=2
0.482962913145
0.836516303738
0.224143868042
-0.129409522551
N=3
.332670552950
.806891509311
.459877502118
-.135011020010
-.085441273882
.035226291882
N=4 (coefficients normalized so that their sum equals 2)
- 0.107148901418
- 0.0419109651251
0.703739068656
1.13665824341
0.421234534204
- 0.140317624179
- 0.0178247014417
0.045570345896
N=5 (coefficients normalized so that their sum equals 2)
0.0386547959548
0.0417468644215
- 0.0553441861166
0.281990696854
1.02305296689
0.89658164838
0.0234789231361
- 0.247951362613
- 0.0298424998687
0.0276321529578
N=6 (coefficients normalized so that their sum equals 2)
0.0217847003266
0.00493661237185
- 0.166863215412
- 0.0683231215866
0.694457972958
1.11389278393
0.477904371333
- 0.102724969862
- 0.0297837512985
0.06325056266
0.00249992209279
- 0.0110318675094
N=7 (coefficients normalized so that their sum equals 2)
0.0037926585342
- 0.0014812259146
- 0.0178704316511
0.043155452582
0.0960147679355
- 0.070078291222
0.0246656594886
0.758162601964
1.08578270981
0.408183939725
- 0.198056706807
- 0.152463871896
0.00567134268574
0.014521394762
N=8 (coefficients normalized so that their sum equals 2)
0.00267279339281
- 0.000428394300246
- 0.0211456865284
0.00538638875377
0.0694904659113
- 0.0384935212634
- 0.0734625087609
0.515398670374
1.09910663054
0.68074534719
- 0.0866536154058
- 0.202648655286
0.0107586117505
0.0448236230437
- 0.000766690896228
- 0.0047834585115
****************************************
LEFT EDGE SCALING FUNCTION COEFFICIENTS
***************************************
N=2
0.848528137424
- 0.529150262213
0.132287565553
0.212132034356
0.838525491562
- 0.484122918276
N=3
0.689047760315
0.724715933318
0.0306457241755
- 0.0291374408035
- 0.986047279405
0.161005005859
0.0172802631418
- 0.0164297845102
0.00443452721664
0.0208960300827
- 0.924034270866
0.380966943247
0.027026368393
- 0.0256962180001
0.0639688879804
0.381971870809
0.176750229211
0.404677356783
- 0.747429432607
0.308155135811
N=4
0.7983434920E+00
0.6022023488E+00
-0.3918024327E-01
0.5194149822E-01
-0.4817281609E+00
0.8739021503E+00
0.1774707150E-01
-0.2352740580E-01
-0.1232594861E+00
-0.6575127688E-01
-0.9620570014E-01
0.9850684416E+00
-0.2636405192E-01
0.3495099166E-01
0.8114147375E+00
0.4440233637E+00
0.3192581817E+00
0.1636579832E+00
-0.4282797155E-01
0.1094933054E+00
-0.1670338745E-01
0.2214378721E-01
-0.1643714751E-01
-0.1112580065E-01
0.2995602574E+00
0.2728668922E-01
0.8472064764E+00
-0.4270166998E+00
-0.3309408518E-01
0.8460780753E-01
0.2727915769E-02
-0.3616415322E-02
-0.5206157868E-01
-0.2836107693E-01
-0.4413123462E-01
-0.1285294872E-01
0.4543141690E+00
0.8282235028E+00
0.3000539798E+00
-0.1037443976E+00
-0.1262470890E-01
0.3227612835E-01
N=5
0.5149588736E+00
0.8572148847E+00
-0.7062608421E-01
0.4242755162E-01
-0.9680855135E+00
0.2366902992E+00
0.1166375834E+00
-0.7006826369E-01
0.2569251462E-01
0.1524482921E+00
-0.8988377800E+00
0.3868800469E+00
0.4678469852E+00
-0.2810519985E+00
0.8265463674E-01
0.5280457443E+00
0.8615307089E-01
-0.2053530811E+00
-0.4422572318E+00
0.4138613016E+00
-0.1341363044E+00
0.8058035557E-01
-0.3810281736E-01
-0.2103133707E+00
0.3026059798E-01
0.2107415482E+00
-0.8553851730E+00
-0.3335028061E+00
-0.1496490635E+00
0.1385649939E+00
-0.2809362553E+00
0.1687682052E+00
-0.4001455131E-01
-0.2777441959E+00
0.9006724682E-02
0.2482896416E+00
-0.9917949547E-01
0.8098971461E+00
-0.2582323890E+00
-0.1241046836E+00
-0.4791583558E-01
0.4436684941E-01
-0.8360649632E-01
0.5022533782E-01
-0.1533783538E-01
-0.9668361515E-01
-0.4952912706E-02
0.6191148032E-01
-0.5197035907E-01
0.2312568194E+00
0.6976664051E+00
0.6350643725E+00
0.1189068293E-01
-0.1727260340E+00
-0.2133407205E-01
0.1975391957E-01
-0.1591308791E-02
0.9559546817E-03
0.3515007474E-03
0.7914817671E-03
-0.1039035431E-02
-0.2096326217E-02
0.2722494187E-01
0.2963704669E-01
-0.3930998378E-01
0.1995579157E+00
0.7233513164E+00
0.6339755574E+00
0.1659420760E-01
-0.1753213559E+00
-0.2110191042E-01
0.1953895348E-01
N=6
0.8035811728E+00
-0.5951951771E+00
0.2409444375E+00
0.3253023901E+00
0.8139624945E+00
-0.4166403612E+00
-0.1713726513E+00
-0.2313725672E+00
0.2850349985E+00
0.2770986033E+00
0.5575844464E+00
-0.6694543662E+00
-0.2876454491E-02
-0.3883540664E-02
-0.5655242446E-01
-0.1151783354E+00
0.5730520088E+00
0.4076173301E+00
-0.4430745754E+00
-0.5409776766E+00
0.9214100804E-01
0.1244008388E+00
0.7306532844E-01
0.2931573652E+00
-0.1321263657E+00
-0.2417688989E-01
-0.3632781234E+00
0.6792063579E-01
0.1887648043E+00
-0.8333645213E+00
-0.1061491551E+00
-0.1433134304E+00
-0.2622879500E+00
-0.6856964195E+00
0.2766563635E+00
-0.8836692218E-01
0.3328884755E+00
0.1288342707E+00
0.5008585553E-01
-0.4619028205E+00
0.8828610310E-02
-0.3895963779E-01
-0.2023876977E-01
-0.2732464068E-01
0.4220601869E-01
0.4941654301E-01
0.3803171587E-01
0.8472554296E-01
0.1946516826E+00
-0.6992838718E-01
0.9438761468E+00
0.1295249721E+00
-0.7015973081E-01
0.1735085659E+00
0.4990121912E-02
-0.2202083175E-01
0.5456011297E-01
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N=7
0.7766665156E+00
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N=8
0.8145313640E+00
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*********************************************************************
LEFT EDGE WAVELET COEFFICIENTS.
*******************************
N=2
- 0.512347538298
- 0.821583836258
0.216506350947
- 0.125
N=3
0.720522830617
- 0.685061028561
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- 0.101939535681,
0.0200641216304
- 0.00827216838973
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0.904072212264
- 0.0510574707719
- 0.18071490589
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- 0.130196008824
N=4
0.5979027428E+00
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N=5
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N=6
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N=7
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0.6530033661E-07
0.6944811113E-07
0.1233981379E-06
-0.7965760361E-07
-0.7731119139E-06
-0.1388121081E-05
-0.5112227575E-06
0.2555383959E-06
0.1026798462E-01
-0.4010231020E-02
-0.1078082259E+00
0.1400472648E+00
0.2886296214E+00
-0.7677643207E+00
0.5361019171E+00
-0.1744125426E-01
-0.4955283493E-01
-0.6789269348E-01
0.3051551317E-01
0.1263630340E-01
-0.1047384889E-02
-0.2681814568E-02
N=8
-0.2620979145E+00
-0.3680051517E+00
0.3007938848E+00
-0.1223236495E+00
-0.3815411480E-01
0.2223806092E+00
-0.7414398574E+00
-0.2856615192E+00
0.7435240137E-01
-0.3958129970E-01
-0.3312367172E-01
0.5296663936E-03
-0.2698631364E-04
0.4827845400E-07
-0.2405836391E-08
-0.1119370789E-07
-0.1530044792E-01
-0.2148297772E-01
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-0.6710033379E-01
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-0.5621677849E+00
-0.2521929541E+00
0.8489402134E-01
-0.5756119710E+00
-0.4252477763E+00
-0.1717046011E-02
0.4224061134E-03
-0.2349772977E-03
-0.4160927634E-03
0.3225252044E-04
0.1266155053E-03
-0.2565139468E-05
-0.1600415263E-04
0.1495099502E+00
0.2099231961E+00
0.1776433332E+00
0.2017476699E+00
0.4784490465E+00
0.5947478107E-01
-0.3620530588E+00
0.6528531425E+00
0.5286410695E-01
-0.2222121460E+00
0.5727745934E-01
-0.1143513897E+00
0.5375754041E-01
0.8506084034E-02
-0.4048226504E-02
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0.5621800279E-03
0.2220046324E-02
-0.4426269435E-04
-0.2761592227E-03
0.2437646831E-01
0.3422639182E-01
-0.4003214950E-01
0.6820587728E-02
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0.3436761211E-01
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0.6694319985E-01
0.4773719174E+00
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0.3636972524E+00
0.5218600648E-01
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-0.4912168611E-01
0.3801081650E-02
0.1492736485E-01
-0.3021310554E-03
-0.1885024820E-02
-0.2526109188E-03
-0.3546846973E-03
0.8694036319E-03
0.1010912744E-03
0.9035494282E-03
0.1885121065E-02
-0.4586723661E-03
-0.3637255134E-03
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0.3593205922E-03
0.3169382593E-01
-0.7649438160E-02
-0.1432336703E+00
0.6121586703E-01
0.4813866596E+00
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0.3644396836E+00
0.5194280857E-01
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0.3808725019E-02
0.1495209647E-01
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0.2622395072E-04
0.3682039585E-04
-0.3484560037E-04
0.1044400200E-04
-0.1346357753E-05
-0.1696820074E-04
0.9452329486E-04
0.1683706766E-04
-0.2973670447E-06
0.1596726796E-04
-0.3381844248E-02
0.5442476019E-03
0.3169178699E-01
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-0.1432957713E+00
0.6127323127E-01
0.4813597833E+00
-0.7771855710E+00
0.3644418761E+00
0.5194577328E-01
-0.2721902826E-01
-0.4913717014E-01
0.3808751988E-02
0.1495225819E-01
-0.3029205141E-03
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0.2304180410E-07
0.3235242306E-07
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0.1082307302E-07
0.4577834066E-08
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0.7406609086E-07
0.2062688911E-07
0.1399229053E-08
0.1155742488E-07
0.3531713891E-08
0.3080661641E-09
-0.3382419079E-02
0.5421367809E-03
0.3169508577E-01
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0.6127335935E-01
0.4813596512E+00
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0.3644418948E+00
0.5194583812E-01
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-0.4913717967E-01
0.3808752014E-02
0.1495225834E-01
-0.3029205147E-03
-0.1889950333E-02
**************************************************************************
RIGHT EDGE SCALING FUNCTION COEFFICIENTS.
*****************************************
N=2
0.848528137424
0.529150262213
- 0.132287565553
0.212132034356
0.838525491562
0.484122918276
N=3
0.889500699418
0.456933808921
- 0.198411021088
0.386241373662
0.832668382346
0.343677222149
0.0390630495665
- 0.0760429831027
- 0.143921176769
0.456707890812
0.807766275151
0.33303120718
- 0.000276799784313
0.000538838660959
0.0349752642807
- 0.085504178774
- 0.134969198046
0.459895243878
0.806892290508
0.332670875027
N=4
0.7629809303E+00
-0.6464209928E+00
0.1555526564E+00
0.1836012627E+00
0.4620817399E+00
-0.8535657052E+00
0.3793246643E+00
0.4477229057E+00
0.4284467089E+00
0.3973740378E+00
-0.2021221018E+00
-0.5228106220E+00
0.2385999808E+00
0.2816233343E+00
0.1056438723E+00
0.1612498770E+00
0.8548427132E+00
0.2929411663E+00
-0.3647382801E-01
-0.9324840384E-01
0.6526723701E-01
0.7703595299E-01
0.7744666349E-01
0.7039069048E-01
-0.6529437593E-01
0.2555397028E+00
0.8099281093E+00
0.4965300820E+00
-0.2995738718E-01
-0.7658857572E-01
0.1517778948E-02
0.1791458518E-02
-0.3127686151E-02
-0.1031248163E-02
0.3237561439E-01
-0.1322822647E-01
-0.9898430026E-01
0.2979273659E+00
0.8037308261E+00
0.4975940939E+00
-0.2963560969E-01
-0.7576592454E-01
N=5
0.9116496405E+00
0.4109682870E+00
-0.1759262671E+00
0.3902566773E+00
0.6301011968E+00
0.6478596732E+00
0.2825379610E-03
-0.6267530579E-03
0.4175827162E+00
-0.4056820940E+00
0.6289312762E+00
0.5152588582E+00
-0.5113299639E-01
0.1134281628E+00
0.4836999717E+00
-0.5526531145E+00
-0.2269223725E+00
-0.5499800740E+00
0.2214856571E+00
0.2051843068E+00
0.2756002147E-02
-0.6113630773E-02
-0.1460819570E+00
0.1465088170E+00
0.1654940540E+00
0.3172833766E-01
0.9500926016E+00
-0.1444971260E-01
0.1180200176E+00
0.1092786192E+00
0.1425700173E-01
-0.3162626148E-01
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0.9173486581E-01
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0.2118550693E+00
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0.9395202801E+00
0.1694709068E+00
-0.1464208651E+00
0.4001747247E-01
0.3705349502E-01
-0.1315787444E-02
0.2918807090E-02
0.1730939444E-01
-0.1895045515E-01
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-0.1556669346E+00
-0.6193750436E-05
0.6330645158E+00
0.7286274296E+00
0.1987062136E+00
-0.3988654553E-01
0.2953429005E-01
0.2734677134E-01
-0.8561071719E-04
0.1899099809E-03
-0.9013424188E-06
-0.1367686226E-03
-0.3599933730E-03
0.3327378221E-03
0.1961899121E-01
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0.1662647625E-01
0.6339762696E+00
0.7234044479E+00
0.1993976399E+00
-0.3913415866E-01
0.2951949191E-01
0.2733306926E-01
N=6
0.6512594203E+00
0.7588551689E+00
-0.6896282953E-01
0.5918480130E-01
-0.9848250307E+00
0.1478544768E+00
-0.1667727153E-01
0.1431265231E-01
0.1090239805E+00
0.7126760308E+00
-0.1719730277E+00
0.6709313982E+00
0.9785571254E-01
-0.8398105097E-01
0.6834310191E-02
0.1247808574E+00
0.7236789907E-01
-0.1108818957E+00
0.1804007431E+00
0.9579551851E+00
-0.5378738739E+00
0.4616103858E+00
0.2733949616E-01
-0.2535533852E+00
0.3484637175E+00
0.3309876136E+00
-0.2950887461E+00
0.1958019515E+00
0.6099724771E-01
0.2691738861E+00
-0.2335808916E+00
0.2004621729E+00
0.5067312783E-01
0.1483313327E+00
0.4193562472E+00
-0.6838768540E-01
0.8092621294E+00
-0.1702431983E+00
-0.5881963586E-01
-0.1104006534E+00
0.5469174586E-02
0.2413483588E-01
0.7905558067E-01
-0.6784653218E-01
-0.2413743535E-02
0.4795434784E-01
-0.2979345169E-01
-0.5477003063E-01
0.1575224560E+00
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0.8600210456E+00
0.4427411664E+00
-0.5350546863E-01
-0.1279990455E+00
0.3964078767E-02
0.1749302183E-01
0.4472356538E-01
-0.3838234812E-01
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0.2792643621E-01
-0.2285669131E-01
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0.4201463682E-01
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0.3288435947E+00
0.7890314114E+00
0.4902682296E+00
-0.4839771981E-01
-0.1181080824E+00
0.3500297801E-02
0.1544640999E-01
-0.2366701878E-04
0.2031134472E-04
-0.3373738591E-03
-0.2266339823E-02
-0.2165041937E-02
0.1906206337E-02
-0.7087323880E-02
0.2020684368E-02
0.4460763770E-01
-0.2107187698E-01
-0.7263524048E-01
0.3379418668E+00
0.7876410397E+00
0.4910540301E+00
-0.4831177914E-01
-0.1179902457E+00
0.3490713064E-02
0.1540411365E-01
-0.1333785556E-05
0.1144672190E-05
0.4933806493E-06
0.2205985224E-05
-0.3259284258E-06
-0.2564526296E-05
-0.1197305244E-05
-0.1010124086E-06
-0.7800395608E-02
0.1767376409E-02
0.4472504864E-01
-0.2106027212E-01
-0.7263753168E-01
0.3379294037E+00
0.7876411418E+00
0.4910559454E+00
-0.4831174259E-01
-0.1179901111E+00
0.3490712084E-02
0.1540410933E-01
N=7
0.7004199839E+00
0.7137309340E+00
0.1053864722E-01
-0.1034210340E-01
0.6658963651E+00
0.7458981223E+00
-0.1757138133E+00
0.1724367832E+00
-0.1665581445E+00
0.1535673633E+00
-0.4422666014E+00
0.8321453685E+00
0.1613128649E+00
-0.1583044098E+00
-0.3601719723E+00
0.3170674120E+00
0.7182714497E+00
0.3180082506E+00
-0.1351100609E+00
0.2887289881E+00
0.2626733923E+00
-0.2577745821E+00
0.5300977359E-01
-0.5460956119E-01
-0.7894206983E-01
0.8761351181E-01
0.1138292645E+00
-0.8840156389E-02
0.1186027472E+00
0.9042968950E+00
-0.9906552530E-01
0.9721797156E-01
0.1575818939E+00
-0.1379327181E+00
0.2550022277E+00
0.1514595280E+00
0.8750776361E+00
0.6499770717E-01
0.2324452664E+00
-0.9349432145E-01
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0.1183889785E+00
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0.6133632819E-01
0.7570194334E-02
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0.2646032111E-01
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0.9366103730E+00
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0.8542222229E-01
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0.4627966328E+00
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0.2151032037E-01
0.4754578984E-01
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0.4829772924E+00
0.2074893977E+00
0.2895141132E+00
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0.1026407474E+00
0.3418617349E+00
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0.3100049663E-01
-0.9439529077E-01
0.9263483604E-01
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0.6801412471E-01
-0.2325217040E-01
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0.4016014208E-01
0.1328240580E+00
0.6204454273E-01
0.1623006055E+00
0.6626531984E+00
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0.2552288166E+00
0.1574205947E+00
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0.1239039332E-01
0.1594142646E+00
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0.7441639637E-02
-0.1106491958E-01
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0.5459386943E-01
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0.2751605400E+00
0.6421729864E+00
0.6324608023E+00
-0.1525303792E-01
-0.8448334026E-01
0.8762658370E-01
0.3318133734E-01
-0.1602650231E-01
-0.1102323809E-02
0.2822484916E-02
-0.6037533035E-02
0.5924934159E-02
0.1534054302E-02
-0.1202064256E-02
0.3182693183E-02
-0.2822206008E-03
0.8251845825E-02
0.6110061754E-02
0.6347460794E-02
0.1773857064E-02
-0.1077767325E+00
-0.1373538162E+00
0.2848100511E+00
0.7687892964E+00
0.5372655282E+00
0.1673410769E-01
-0.4968993169E-01
0.6802066194E-01
0.3052110650E-01
-0.1264613714E-01
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0.2681998325E-02
-0.2814830317E-04
0.2762334252E-04
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0.2750781612E-04
0.1564675966E-04
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0.2143345853E-04
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0.2578289227E-04
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0.1744136474E-01
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0.6789267353E-01
0.3051551282E-01
-0.1263630236E-01
-0.1047384891E-02
0.2681814574E-02
N=8
0.8154844606E+00
0.5787789687E+00
-0.1247831717E+00
0.1758162321E+00
-0.3656686728E+00
0.9054304139E+00
0.2997291603E-02
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0.1242773642E-01
0.6252204333E-02
-0.9904302529E+00
0.1372135453E+00
0.1966225161E-01
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0.3499437320E-01
0.3331775846E+00
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0.4323222764E-02
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0.7954572256E-01
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0.9561140215E+00
0.2524537323E+00
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0.1516837508E-01
0.1099880629E+00
0.1862464672E-01
0.1115880939E+00
0.2254220157E-01
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0.4998095408E+00
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*************************************************************************
RIGHT SIDE WAVELET COEFFICIENTS.
********************************
N=2
0.512347538298
- 0.821583836258
0.216506350946
0.125
N=3
- 0.409742284317
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N=4
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N=5
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N=6
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N=7
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