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RDA147 386 THE STABLE EVALUATION OF MULTIVARIATE BSPLINESU Ili


WISCONSIN UNIV-MRDISON MATHEMATICS RESEARCH CENTERT A GRRNDINE SEP 84 MRC-TSR-2744 DRRG29-88-C-884iUNCLASSIFIED F/G 12/1 NL lEhE hhhELa -Le 0NRC Technical Summary Report 2744THE STABLE EVALUATION OF

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Document on Subject : "RDA147 386 THE STABLE EVALUATION OF MULTIVARIATE BSPLINESU Ili"— Transcript:

1 RD-A147 386 THE STABLE EVALUATION OF MUL
RD-A147 386 THE STABLE EVALUATION OF MULTIVARIATE B-SPLINES(U) Ili WISCONSIN UNIV-MRDISON MATHEMATICS RESEARCH CENTER T A GRRNDINE SEP 84 MRC-TSR-2744 DRRG29-88-C-884i UNCLASSIFIED F/G 12/1 NL * lEhE hhhE La.. * .- Le 0 NRC Technical Summary Report # 2744 THE STABLE EVALUATION OF MULTIVARIATE B-SPLINES (.0 Thomas A. Grandine lit 4 0 Uneivet ofus Wiscnsi-Maiso LELm Approved for public release Distribution unlimited Sponsored by U. S. Army Research Office P. 0. Box 12211 84 1 e 3 Research Triangle Park 1 6 23 North Carolina 27709 .... UNIVERSITY OF WISCONSIN -MADISON MATHEMATICS RESEARCH CENTER THE STABLE EVALUATION OF MULTIVARIATE B-SPLINES Thomas A. Grandno Technical Summary Report #2744 September 1984 /~% .J". -4. ABSTRACT This paper gives a general method for the stable evaluation of... multivariate B-splines. The problem of evaluation along mesh boundaries in discussed in detail. Several examples are presented to demonstrate the effectiveness of the method for arbitrary B-splines. -, AMS(MOS) Subject Clas fication: 41A15, 65D07 Key Words: B-spline, implex spline, recurrence relation, linear progran simplex method. Work Unit Number 3 -Numerical Analysis and Scientific Computing Accession 7*1" NTI.S GRA&Z -- DTIC

2 TAB Unannounced ] JUstifioatio Distribut
TAB Unannounced ] JUstifioatio Distribution/ "... __ -.,,.,. .4.'. Availability Codes .t.: ri and/or .4 Dit, 300olal Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. .4... .. a",'.''":.:.;...'. -.:o.". .;"; ' " '.' '. ';.,.' ' -''". .,.,.". .,'' '.' ',' ' ',.' .. ., ..".,.- ,,'..,.. , I "o";- -,.'.',*Sponsored''.-:,"" , "l. .byteUie States % ," ..; ". Army nderContrct N.,.",", .DAG2-O-,-0 I. .",' , "". .'"' ,.." |' , ,/,,,,. iZ,.,o ,,_',. 'L,.'J .-,,_. ..'...,...;.'.. ', ', .., ,,:.. :,,,...,,,. ., .-.,.....,-.. ,.... ...,.-.:-....',.,... SIGNIFICANCE AND EXPLANATION For one variable, the problem of stably evaluating Ef-splinex via their -rer ce relations is well understood. For multivariate 3-splines, however, the geometry becomes more complex, and it becomes quite difficult to implement the recurrence relations in such a way that one ends up with a robust evaluation method. This paper presents a method which guarantees the stable evaluation of all th multivariate simplex B-splines. I I ' The responsibility 'for the wording and view* expressed in this descriptive. summary lies with K1C, and not with the author of this report. N: .% ., . _,, .. .. , . t ", .., ..5, ,'.. .., ., .' ..0 ..

3 % ... .. 0 ,. .*. 5 ' ..' .'... .....
% ... .. 0 ,. .*. 5 ' ..' .'... ...... .......... .....~ '"4 --7--- .00.) bok*. WO pox) 7%w L bibs.his .at thtigow A bft&* gri A a. a~m ~hs- t .semsv~~u~ INI C A Ml owS 4 1 a"lin NowS ata iI&Mt~SDpsa~&~S.C(' 104^ Wit =Qi Psi tht Iu#prot ofU1Sm WeIM fors hug siveni wat byplm a nubrfpat'. is"Inc twM (I", 4Pe4, winth RAW 1," WW.hen 6 ad TV& or ak spoof olethion ho~a" hsoe.pni aiu ormb ub, epe SAMvan Tihoe 1, Wit ahcoop epl to ie M siftle (S- shmshoe ~ 5h Sthded ian ade Aoul'mg id n alut pevus cAMatolm s "(hc mfn to .Minte uadiowwoul P &to ae') m aidciey ~t a ut.u OW howit in" sk quedie feht 2: 0o, s soi. a. ~fe ta1TwArdi sk.thet Wlc arMT .. whias ooe bh. tu waite weh i&d_ to eaUa Soapbu to lu (3) M "o e"ft oleti The ifilculy vt~h ge aU 0,ftotdiy cOnotsn Milmon dlctatu that anot plbbo d a be mm. Thu, ye writiletn t o radi a -azlat the ofltowing two sprvie --a- 404 240 which is not a solution to (5). Fortunately, there is a well-established procedure for handling problems of a similar nature. Except for the absense of an objective function, (5) is of the same form as a llnear prognmming problem. Such problems have been studied in great detail over the years, and many ways of coin- puting their solutions are known. Th

4 us, an approach which one might consider
us, an approach which one might consider is the introduction of a "dummy' objective function to convert (5) to a linear programming problem which may then be solved by standard techniques. In practice, this seems to work quite well. In the numerical experiments performed to date, the simplex method, implemented with the help of the Tucker tableau (described in the Appendix), has performed admirably. One introduces the objective function 0 (for reasons to be made clear shortly) and treats (5) as a linear programming problem; i.e., one considers the following problem, equivalent to (5): max 0 subject to xaiPt, =X i=O (6) iWO 0i _0 n =,..Y Since the objective function is constant, solving this problem amounts to finding a feasible point. In other words, it amounts to finding a solution to (5). In solving (6) via the method outlined in the preceding paragraph, one can see why the particular choice of an objective function is, in some sense, optimal. In the Tucker tableau, the pivot rules are such that a row of all seres will never change. Thus, the physical storage of the row corresponding to the objective function in memory is unnecessary. Furthermore, the dual of the problem is given by min (z,u) + subject to (Pti, u

5 )+ �20 , i=0,...,n. This problem
)+ �20 , i=0,...,n. This problem is dual feasible in that u = 0, v = 0 satisfies the constraints for this problem. This situation arises because the objective function of the primal problem (6) has non-positive coefficients. This means that (6) is *easy to solve because one can dispense with phase I of the simplex method and use instead the dual simplex method. Example 4: Consider again Example 2. Setting up this problem as a linear programming problem leads to the tableau 00 01 02 a 0 *4 0 , ro (-1 -1 0 1 1 0 r, 0 -1 -1 0 1 1 -4 r2 -1 -1 -1 -1 -1 _-1 -1 where the variables ro, rl, and rl are so-called Islack' variables. Since (6) is made up of equality constraints, one first pivots ro, r,, and rs to the top of the tableau and, once this is done, deletes the columns corresponding to them. After all, the variables along the top are assumed to be 3 M~~~ MO SAV K Ir - zero, and the deletion of a column corresponding to such a variable merely makes this condition permanent. Thus, one first exchanges ro and ao to get rO 01 02 03 014 C~IS 0 -1 - r, 0 -! -1 0 1 1 pr -1 0 -1 -2 -2 -I Now one deletes the first column and exchanges r, and a, to get Oc rl 0 -2 03 04 06 1 01 -1 1 0 -1 -1 4 ,, 0 -1 -2 -2 -4 Lastly, one d

6 eletes the column corresponding to ri, e
eletes the column corresponding to ri, exchanges r2 and a2, and deletes the resulting column corresponding to r2 to obtain 0 1 2 2 1 0 4 0~-2 -3 -2 4.. Q2 2 2 1 14") This s not (dual) optimal, so one must exchange a, with some column whose entry in a's aw is. negative. Since all columns satisfy this, the first is chosen, and a and as are excheasd to reveal 01 C04 05 1 C 2 2 C3 2 2 %. . 02 1-1 -1 1 This is a solution to (6), and therefore to (5). Note that although not explicitly required, only 3, or m + I of the ai are non-zero. This occurs because (6) only had 3 constraints (other than the ." non-negativity constraints), and solutions of a linear programming problem must satisfy a corn- plementarity condition; that is, the only variables which can be non-zero are those corresponding to tight constraints. Since r0, ri, and r2 were forced to be zero, there can be at most 3 non-zero a,. Thus, the linear programming approach implicitly takes care of the efficiency issue discussed above. *. Once one has solved (6) by this approach, one can eaily evaluate the spline using (4), assuming that the values of the lower order splines which occur on the right hand side of the equality are known. In general, this is not the cas

7 e, but one can reapply the technique to
e, but one can reapply the technique to each of the splines . appearing on the right hand side of (4). This process may be carried out inductively until one can finally express the value of the desired B-spline at the desired point in terms of piecewise constant functions at that point. The Tucker tableau makes this inductive process extremely efficient. The tableau which solves , (6) may be used to solve the resulting subproblems. One can view each of the subproblems as , being just (5) with the additional constraint a, = 0. Thus, to solve a subproblem, one can take 4 e %. . -I. -.-, r " -, .. ....,..,., , .... ., ' , ., ... , ., , ,;.",' '''," .,.....,.'. ," .,,,." ' ' :, the tableau which solves the main problem, pivot a, to the top of the tableau, delete the column ! : corresponding to it, and then optimize. This will yield a solution to the subproblem in short order, often only one pivot. These solution tableaus for the subproblems may then be used to obtain cheap solutions to the sub-subproblemas, etc. This exploitation of similarity among the various linear programming problems which one must solve saves an enormous amount of work. At first glance, it would seem that the difficulties in computing the value

8 of a multivariate B- spline have been o
of a multivariate B- spline have been overcome. Unfortunately, one of the more persistent of the problems has yet to be overcome. The piecewise constant functions which one ultimately ends up with have discontinuities along certain boundaries, namely along the grid lunes. A grid line is a set consisting of the convex hull of m or fewer points taken from the set {Pto, ..., Pt,,). A point z is said to lie on a grid line 10 if it is a member of some such set. Whenever one wishes to evaluate a spline at a point lying on such a grid line, one runs the risk of computing it improperly. Example 3: Suppose one wishes to evaluate M(0,Oto,ti,t2,ts), where to = (1,1,0), ti = (-1,1,0), t2 -(-1,-1,0), and t3 = (1,-1,1). Then (0,0) = IPto + lPt2, and therefore f(0,O1tot14t4t) = M(0,01t1,t + 3-M(0,O1tot1,ts). But M(.It,t3,ts) and M(.Ito,t1,t3) are discontinuous at (0,0), so it is unclear whether to choose the interior or exterior limits as values for these splines. If interior limits are chosen, the computed value of the spline will be twice as great as the actual value. If exterior limits are chosen, the value will be zero, and this is obviously also incQrrect. There are many ways of attempting to circumvent this problem, but ne

9 arly all fail in some way, especially wh
arly all fail in some way, especially when one takes into account the inexact nature of the arithmetic performed by the computer. In general, there seems to be no reasonable way to handle this problem, but for smooth splines, there are a few things one might try. The obvious approach is to prohibit one from evaluating a spline on the grid lines. This is certainly the most sure-fire answer, and it is also a simple enough scheme to be easily implemented. . All one need do whenever one finds that he is on a grid line is to move the point in some direction S.... by e and try again. Higher order splines are, in general, continuous, so this small change in the -. location of the point will make a very small change in the value of the spline. Unfortunately, this . must be done by hand, since the computer is unable to tell when a point is actually "on' a grid line; the best it can usually do is to tell when a point is "near" a grid line. As the number of variables increases, the structure of the grid lines becomes increasingly complex. As a result, it L becomes increasingly difficult to avoid computing the value of the spline there. For example, in one variable, the grid lines consist only of the knots, while in two variab

10 les, the grid lines consist of the knots
les, the grid lines consist of the knots as well as the line segments joining the knots (see Figure 1). In one variable, the problem isn't so terrible. Typically, one decides that all the piecewise constant splines are either continuous from the left or continuous from the right. Then, when one ,-.. needs to evaluate at a knot, one sets the value of the spline to zero if it is the left knot, for example, and non-zero if it is the right knot. Thus, for one variable at least, little needs to be done to clear up this nuisance. Conveniently, such a plan of attack generalizes to more than one variable. One merely chooses (somewhat arbitrarily) some direction in RM and evaluates piecewise constant splines according to the following rule: If z is in the interior of the region of support, or if z is on the boundary of the region of support and the arbitrarily chosen direction points into the interior, the value of the piecewise constant spline shall be used; in all other cases the value of the spline shall be 0. In theory, this rule eliminates the difficulty. Unfortunately, because of roundoff error, one can have both situations occurring, and the ambiguity about what to do persists. Thus, for, tb, 55 V * S.. * % ~ ~ ' % %

11 .5 multivariate splines an alternate ap
.5 multivariate splines an alternate approach must be taken. Its successful implementation depends on the following theorem. Theorem 2: Let to, ...,t,, 1 be a collection of points in general position in R". Let A [to,...,t,,1 , and let A,:= [to,...,9 1 ...,t,,t.+. Then for all x : A, with : not on any of the grid lines, x lies in exactly two of the Ai. Proof: The statement z E A is equivalent to the statement that there exists a solution to the problem: nt+l E giti --= X"':" i=0 n+ 1 (7) �a,0, i=o,...n + 1. Ignoring for the moment the inequality constraint, one can view (7) as a linear system, namely where T is an n + 1 x n + 2 matrix, a is the vector whose individual components are the ao, and P is the vector obtained by adding the component I to the end of x. T clearly has rank n + 1, since the ti are in general position. Thus, this linear system has a one-parameter family of solutions, say .... a(s) : + sx, where V, a E R"+ I and 8 E R. Now one can consider the inequality constraints, ;.. .._ ai �_ 0, i = 0,...,n + 1. This is equivalent to V + szi 2! 0, i = 0,..., n + 1. Taken together, all these conditions define some interval S := Is-,s+) in which s must lie in order for a(s) to satisfy the ineq

12 uality constraints. Since x E A, it is c
uality constraints. Since x E A, it is clear that S is non-empty. Furthermore, it is clear that = 1 d = si = 0, for E"+= a, = 1, independent of .. Since the solution cannot be unique, at least one of the zi is non-zero. But E"_+1 z = 0, so there must be at least two of the z. non-zero and of opposite sign. When zi is positive, one gets a lower bound for s, while z, negative gives an upper bound for s. Hence, S is a finite interval. Since a(s) is a continuous function of L 'Y a, no components of a can have sign changes in S. Furthermore, s outside of S means that one or more components of a(s) are negative there, hence must change sign on the boundary of S. Suppose ai(s) = 0 and a.(s+) = 0. Since x does not lie on a grid line, a,(s_) is the only sero component of a at s-. Similarly, a,(s+) is the only zero component of a at e+. But this says that x E A. and : E A,. Since a(*) is aine, no other solutions with a zero component are possible. Thus, : lies in exactly two of the A. This proves the theorem. With this theorem, one can now correctly evaluate continuous piecewise linear B-splines. Suppose one wishes to evaluate M(zlto, ...,t n+), where Pto, ..., +1 are points in general position in R". Then, after one solves

13 (5) to get :.. nt+1 : M(Zlto, ...,t.+i)
(5) to get :.. nt+1 : M(Zlto, ...,t.+i) =(n + 1) 1: &,M(Zlt0,.,,ot+,.,t, ,' ", 1=0 it is clear that if x does not lie on a grid line, then all but one of the M(l0to,..., t it., 1,..., t ,+I) will be zero. This is an immediate consequence of Theorem 2. If : is ou a grid line, however, one can still impose this condition. This trick forces evaluation on the grid lines to behave just ?. like evaluation off the grid lines. One must be careful, however, to coose the correct piecewise ,. constant spline to be non-zero. 6 %. ' d Z ~j- % %. Mspo lst C*Amwsi again Exaaipls 4. Oeiposes the condition that exactly one of A(.~,s~e. mni ~ej.,,, .) ambe nM-aI." at (0,0). In this Cas, it doesn'tmae? rwhat .me chasm Nomae on. nigh attempt to sol" this problem numerically with the follwing resut: M(,jo~ztit)=M(0,Ofaj~2,,,) + 10 MU(0,O1taot,,a) + (,4,1t) one clearly cannot choos M(oIts, t2, te) as the only non-sneo spline and expect to get reas"bl Thbis Indiate thato.. unt be careful in the selection of the non-nrc spline. A good method is to choose ficom all possible spline, the one which has the largest coefficient. This will eliminate the kind ofawwumric nuisance which occurs in Example S. However, it hot e pointed out that this a

14 pproach only works if the linear spline
pproach only works if the linear spline is cown timun. If it is disc"otinuous, onse can evaluat the spine staby everywhere excep slong the disontnuty, where numeric noise makes the exact locaton of the discontinuity impossible to cal- culat. Fartunsiely, for smooth splines at least, this never happens, and one needn't he concerned With it. Gives that one can evaluate continuou linear splines stably everywhere, one can evaluate smth higher depes spines saly everywhere. Instead of expressing the higher degree, spine as a BMm combination of contalat -splines, one stops oae level sooner OWd expressee it as a linear combnatoin aer splines, eac of which is continuous and can be evaluated stably by the method Just 4escribed. One no longe worries about extraneous term resulting frm grid line efft because the linear wpoe are continuous. Using "hi technique, many difernt splines have been computed in the bivariate case. Some * .1o these appear on the follwing- pages. In order to produce these graphs, the mesh was deliberately chosen so that many evaluations, along grid line were necessary. The knots used hr ach spine are give at the bottom of each paop. Linear Bivariate B-spline- 1 0 Quadratic Bivariate B-spline * I * I * ,*

15 . -- I,. * I * I r * I I. *.. %. 'I V *
. -- I,. * I * I r * I I. *.. %. 'I V * I S * I * . * (1,O),(O,1),(-io),(o,-i),(ii) 9 C * .*.. ~* **** * * -. -, * .* * -*.* * * .,. .** * .* * .-* * -. * -*.* -. .*-.*..*. *. .*..**............*.*.*.*.*~%*.-*.%*. .*.*..**--.* '-1 ........................*.*.*.................. **%* ~ ..* * .* * * .- ....................... *.....* LJAVOiI~L~ - '~ I.., ~- '. 4 ~ ~ 9% d ~*.. .,. 66 * * ~%* .9 .9. -9 ____________________________________ L. Sb 9. h %II ~ 9 9 ' *,.... ~ ~ .~ -------- Degree 7 Bivariate B-spline ~,.. ~ q' V. * 1* L~*. A 0 9% .9- * "-9 9~' 9. -p *~ ~ I, (sin 2irk/1Ocos 2Trk/1O), k=1,...,1O xi T '--~ -- '~W&'~% Appendix: The Tackw Tableau Consider the following linear programming problemn: min z:= (c,z)-d subject to r := b- Az _ 0 (Al) �z._, i1,...,a. Each constraint, r_ 0 or zi �_ 0, defines a half-space in which the solution must lie. Since one must satisfy all the constraints, the solution must lie in the intersection of these various half-spaces, i.e., z must lie in some polyhedral region in RO, the feasible region. The functional which one wishes to minimize over this region is linear, so the solution must lie at an extreme point of the easible region, sometimes called a vetex. A vertex

16 is, in general, determined by specifyin
is, in general, determined by specifying a of the bounding hyperplanes to which it belongs. This means that a vertex is determined by setting n of the numbers ri, r2, ..., rm, st, z2, ..., zn to sero. These re the omn-basic variables for that vertex; call them G, C .. The remaining variables are the bask variables for that vertex, and they will be denoted by Pl, P2, ..., Pro. They are related by the equation p = p TC, (A) which is obtained from the equation = b-As (AS) by partial Gaus-Jordan elimination, i.e., by solving (AS) for pa, p2, ..., p.. Initially, C = z and p = r, i.e., one is at the vertexa z = .The vertex specifid is feasible exactly when P 2! 0 (since that makes all the basic variables non-negative while the non-basic ones e ero by choice). In order to keep track of the value of the objective functiom, z, one expresses it always in terms of the non-bsic variables, 8= ( -5. (A4) Initially, 7 = c and 6= d. For the computations, one keeps the msential hn ation in the Taclsr tableau, as follows: -C6 -C2 -C. 1 -p -s... -15 1i P ] ill t2 ti A ) *1* :~ I (AS) Pin /tft ti .tns tin. S 11 73 ... 7. The simplex method operates on this tableau in the following way: if each entry in the Anal column is non-negati

17 ve, thean one is currently located at a
ve, thean one is currently located at a feasible vertex. One wishes to move to a neighboring vertex where the value of the objective function will be less. If one of the entries in the fnal row is negative, then one can increase the value of the corresponding non-basic variable and reduce the size of the objective function. This is done by exchaninsg the non-bmic vaiable with a basic variable. This amounts to solving the equation representing the basic variable for the non-basic variable and then substituting the resulting expression into all of the other equations. Suppose one wishes to exchange the variable in the k-th row with the variable in the I-th column. The following pivot rules can be derived to carry out this proces. Let T represent the 12 entire tableau, and til its i,j entry. Then t --G/tkl i = 1,...,k -1, k + 1,...,m + I (A6) tkj -- t /t~t j = 1,...,! -1,1+ l,...,n + 1 :. --... ti3 "- ti, -tat h/t' . In order to maintain feasibility, one must choose the pivot row carefully. This is done by selecting the row k by the following rule: Choose k so that -m i t, , � 0 (AT) ti,1." t j .. If more than one row satisfies this, then any of them may be chosen. One performs exchanges until the final row of

18 the tableau is non-negative. One then ha
the tableau is non-negative. One then has an optimal solution. Of course, if the rightmost column has some negative entries, then there is some extra work involved before one can start this optimization process; one must first find a feasible vertex. For- tunately, the early pioneers of linear programming noted that the simplex method can even be used to tackle this problem. After all, it is a procedure which transforms a tableau into one whose last row is non-negative, while preserving the non-negativity of the left column. So, one simply considers the dual simplex method, in which the roles of basic and non-basic variables are re- versed. A new objective function (0 works well because it eliminates the necessity of adding a row to the tableau) is added to the problem. This objective function should have the property that its coefficients in terms of the current non-basic variables are non-negative, but no other restrictions need to be placed on its selection. Now one has a tableau whose last row is non-negative, but whose last column does not share this property. One can apply the dual simplex method to this tableau to get one in which both the last row and the last column are non-negative. The dual simplex metho

19 d proceeds just as the simplex method pr
d proceeds just as the simplex method proceeds, except that the row and column pivot rules are interchanged, the former pivot column selection rule is now used to choose the pivot row, and the column I is chosen by the following rule: ChooselIso that -i -tM1,. (g One continues exchanging basic and non-basic variables until the rightmost column is non-negative. At this point, one has a feasible vertex, and the new objective function can be replaced by the original, which, of course, must be expressed in terms of the current non-basic variables, something best accomplished by keeping track of it all along. One can now proceed with the optimization process as before. , This entire procedure, as well as many of its alternatives, is described in Dantzig t19631. . Mangasarian [1978] discusses the Tucker tableau and many of its applications...' " ..-.", *., ._** * 13 P.' ,*% % , ,. --......-... ' ..., .. ..... . ...,. [I) do Dw, Carl 11W91. 49pm m aer ambinslms df DepHin App -u-@g e S "2m A, G. G. Loadst, C. K. (bmi, md L. L. Scbmmaker, o.,1 Acadmki Prom, pp. 1-4?. [2) do Doar, Card (19"2. wropis in MUtiwistie ApoitonTheowy' m. lboe is Noewi. Amedphde P. 1im, ad. rhsWr Lectur Notes in M~taide 90, IM6, Mp WWTI 131 do Boor

20 , Carl, and SM&g Kim.. (1961). 4ecm~me R
, Carl, and SM&g Kim.. (1961). 4ecm~me Rulations 6r MulStvrilm &WNWm1, Mathundties Rummach Costsr TSR #2215. Pvmc. Amer. Me&h Sem., to appear. 141 do Door, Carl, md Slif, 3(1m 11982]. D-.pbsn fram Palm ipe&,'- Mabomadti RA. march Centr TSR #2=20 [5jl Danteig, G. B. 11963. Limaw Prgmming ad Iutnaie.. Prbrnsom Ulvuhp PraL [6] Hakooan, H. (IMa) "On Multirat Depls, in SUAM harmd of Newe" Amdpui. i9., pp. 510-517. 17] HaW5, 3(1m. [1O81J. 6A Remark an Muitivrist. Epla., in Jeurnal of App -A Wk rhom*p33, pp. 119-125. 181 HRllg, 3(1m. [1962]. "Multiverias. SpHi., in SIAM Jered ef NvmcrikeAmeif.10., pp. 1013-1031. 191 M&OAngmian, OMv (1978]. -UROger Propuumiag Lecture Notes.' [101 Mlechuli, C. A. 119191. *On a Numerically tgimnt Mstod hr Computing Muldhvaris 3- Rplf.,' in Mmbinrkkt Approximation "w"er, W. Schamp and K. Zefle, e06., WNM 51, 144 SECURITY CLASSIFICATION OF THIS PACE (Iho. Data XROOM REPORT DUMAENTATION PAGE VA M~M 1.RPR RSUM 2. GOV? ACCESSION NO' .IPIET S CATALOG NUMBMER 2744 i 4. TITLE (M S.ilo)5 TYPE OF REPORT 6 PERIOD COVERED *'h Stable Evaluation of multivariate 3-Splines Sumryeport peroseii 6. PERFORMING ORG. REPORT NUMBER .*. J. AUTN01o) .R(TIA" 01AT pB Thas A. Grandin. DAAG9-S-C-0041 11. PERFGRMING ORGA

21 NIZATION NAME AND ADDRESS 0P Mathematics
NIZATION NAME AND ADDRESS 0P Mathematics Research Center, University of Al.tE~I1jTS Work 10nit Number 3- 610 Walnut Street Wisconsin NmrclAayi n Madison. Wisconsin 53706 111. CONTRDLLING OFFICE NARM AND ADDRESS I&. REPORT OATE U. S. Army Research Office September 1964 P.O. BOX 12211 N umBao or PAGES9 Research Trianie Park, North Carolina 27709 1A- 14. MONITORING AGENCY SME a ADDRE5S'* dIUMMI ben OOMNOIb Ol~lee) IS. SECURITY CLASS. Wo Wei 'rpo UNCLASSIFIED 15. ~k~JIFCATION/DOWNGRADING IS. DISTRIBUTION STATEMENT (of Me Rp" Approved for public release; distribution unlimited. 17. DISTRIGUTION STATEMENT (of No. a&*~ tered to Sloe* So. NI 49toe A opin) pav IS. SUPPLEMENTARY NOTES a.. B-spline, simplex spline, multIvariate, recurrence relation, linear programming, simplex method. 29. ABSTRACT (Casntho an eves oai ode HI noo..m and ldeniFy byF blook nunm.) *This pgper gives a general method for the stable evaluation of multivariate 3-splines. The problem of evaluation along mesh boundaries is discussed in h.N * detail. Several examples are presented to demonstrate the effectiveness of the method for arbitrary B-splines. DO JA Un47 EDITION OF I NOV 65 IS OBSOLETE NLSSFE SECURITY CLASSIFICATION OF IHIS PAGE (?07m 7Dato ROe.