<content> GaussLegendre Coefficients </content>

63f46292-e610-43aa-aefc-3febf4ab985f 14 Alvaro Diaz Falconi 4 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

x n LegendrePoly n 0 ≡ 1 n 1 ≡ x x x n 1 - LegendrePoly * n 1 - (x n 2 - LegendrePoly * - (if if 11 line :

x n LegendrePoly' n 0 ≡ 0 n x x n LegendrePoly * x n 1 - LegendrePoly - x 2^1 - / * if 11 line :

n TOL LegendreRoots n 2 < 01 matrix R 0 : k 1 0.5 n * Floor range x π k 0.25 - n 0.5 + / * cos eval : Iter 1300 range xo x : x x x n LegendrePoly x n LegendrePoly' / - eval : x xo - abs TOL < break continue if 31 line for R k el x : 31 line for n 2 mod 0 ≡ R - R stack sort eval R - 0 R stack sort eval if 31 line if 11 line :

61012 -^LegendreRoots 0.999999999999591 - 0.616706590193043 - 0.616706590192188 - 0.616706590192188 0.616706590193043 0.999999999999591 61 mat

Works, but very slow.

Coeffs from:

https://keisan.casio.com/exec/system/1280801905

GL.20 1 - 0.005263157894736842105263 0.9807437048939141719255 - 0.03223712318848894149161 0.9359344988126654357162 - 0.0571818021275668260048 0.8668779780899501413099 - 0.0806317639961196031448 0.7753682609520558704143 - 0.101991499699450815684 0.6637764022903112898464 - 0.1207092276286747250994 0.5349928640318862616481 - 0.1363004823587241844898 0.3923531837139092993865 - 0.1483615540709168258147 0.239551705922986495182 - 0.1565801026474754871582 0.0805459372388218379759 - 0.160743286387845749008 0.0805459372388218379759 0.1607432863878457490077 0.239551705922986495182 0.156580102647475487158 0.3923531837139092993865 0.148361554070916825815 0.5349928640318862616481 0.1363004823587241844898 0.6637764022903112898464 0.120709227628674725099 0.7753682609520558704143 0.1019914996994508156838 0.8668779780899501413099 0.080631763996119603145 0.9359344988126654357162 0.057181802127566826005 0.9807437048939141719255 0.032237123188488941492 1 0.005263157894736842105263 202 mat :

GL.50 1 - 0.0008163265306122448979592 0.997005175362643586236 - 0.0050273455497416577264 0.989972220058020802022 - 0.0090329866868996020993 0.9789519106467198092673 - 0.013000139812211760657 0.9639894801099686121339 - 0.0169146554838637660232 0.9451453286808814137682 - 0.0207609894970159934393 0.922495378502752746657 - 0.0245237017181694943149 0.8961308490768326327616 - 0.0281876554510602491663 0.8661579081773533109677 - 0.0317381026065706387509 0.832697249796457952787 - 0.03516074908934724962954 0.795883610117728469387 - 0.0384418142109977345625 0.7558652257940530390604 - 0.041568086852368492733 0.7128032374236458598544 - 0.04452697893447274223 0.6668710408790448158956 - 0.04730657622923672285351 0.6182535892015291708338 - 0.04989568639334337604645 0.5671466479156704257289 - 0.052283884066433463416 0.5137560067830487440676 - 0.054461552866930615293 0.4582966511792997592748 - 0.0564199241232101410276 0.4009918964363984808992 - 0.058151112187474100161 0.3420724886388038436291 - 0.0596481461918334541467 0.281775675495909512023 - 0.060904998119632791003 0.2203442510330564760657 - 0.0619166070794861790183 0.1580255779484029029865 - 0.06267889968456801074686 0.0950705915726566477616 - 0.0631888064552666332142 0.0317327894426237129119 - 0.0634442741792528868443 0.0317327894426237129119 0.0634442741792528868443 0.0950705915726566477616 0.0631888064552666332142 0.158025577948402902987 0.0626788996845680107469 0.220344251033056476066 0.061916607079486179018 0.281775675495909512023 0.0609049981196327910029 0.342072488638803843629 0.059648146191833454147 0.4009918964363984808992 0.0581511121874741001609 0.458296651179299759275 0.0564199241232101410276 0.5137560067830487440676 0.054461552866930615293 0.5671466479156704257289 0.0522838840664334634156 0.6182535892015291708338 0.049895686393343376046 0.6668710408790448158956 0.047306576229236722854 0.7128032374236458598544 0.04452697893447274223 0.7558652257940530390604 0.041568086852368492733 0.795883610117728469387 0.0384418142109977345625 0.832697249796457952787 0.0351607490893472496295 0.8661579081773533109677 0.0317381026065706387509 0.8961308490768326327616 0.0281876554510602491663 0.922495378502752746657 0.024523701718169494315 0.9451453286808814137682 0.0207609894970159934393 0.9639894801099686121339 0.016914655483863766023 0.9789519106467198092673 0.013000139812211760657 0.9899722200580208020225 0.0090329866868996020993 0.9970051753626435862362 0.005027345549741657726 1 0.0008163265306122448979592 502 mat :

2 f a b c d GL DInt.GL Cx a b + 2 / : hx b a - 2 / : X Cx hx Cx UnitsOf * GL 1 col * + eval : 13 mat Cy c d + 2 / : hy d c - 2 / : Y Cy hy Cy UnitsOf * GL 1 col * + eval : 13 mat N GL rows : m n Clear 12 mat hy hx * GL m 2 el GL n 2 el X n el Y m el f * n 1 N sum * (m 1 N sum * 41 line :

Determine if the point (x,y) is inside or outside the Polygon P P x# y# InPoly I 0 : c 1 P rows 1 - (range s x# P c 1 el ≥ (x# P c 1 + 1 el > (& (x# P c 1 el ≤ (x# P c 1 + 1 el < (& (| (: I I s 0 ≡ (P c 2 el P c 1 + 2 el - (x# P c 1 + 1 el - (* y# P c 1 + 2 el - (P c 1 el P c 1 + 1 el - (* > (& + : 21 line for I 2 mod 31 line :

Pseudo random number between 0 and 1 rnd 109 -^109^random * 11 line :

Pseudo random number between a and b a b rnd a rnd b a - (* + 11 line :

Poly 0.5 - 1.5 - 0.5 - 0.5 1.5 - 0.5 1.5 - 1.5 1.5 1.5 2 0.5 0.5 0.5 0.5 1.5 - 0.5 - 1.5 - 92 mat :

P 0.5 - 1.0 0.4 1.1 - 0.7 0.8 - 0.2 - 0.4 1.5 - 0.2 - 0.7 - 0.8 - 0.5 - 1.0 72 mat :

a b c d 22 mat P 1 col min P 1 col max P 2 col min P 2 col max 22 mat eval :

r 11000 range :

X r el a b rnd :

Y r el c d rnd :

P X Y . 8 clr r el P X r el Y r el InPoly 1 ≡ red blue if : augment 2 1 sys

x y f x 2^y 2^+ (P x y InPoly * :

x y f a b c d GL.50 DInt.GL 3.6329

x y f x a b int y c d int 3.6274 ≈

Alvaro