Computer Programming

Transcription

1 Computer Progrmmg I progrmmg, t s ot eough to be vetve d geous. Oe lso eeds to be dscpled d cotrolled order ot be become etgled oe's ow completes. Hrl D. Mlls, Forwrd to Progrmmg Proverbs b Her F. Ledgrd T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş

2 Outle Smple progrms pseudocode Computto of the determt d verse of mtr Appromto of fucto b poloml of degree m usg lest squres method Numercl tegrto of the equto =f () wth tl codto ( 0 )= 0 Solvg ler equtos For ll smples: Problem defto Mth model Algorthm developmet T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 2

3 Computto of the determt d verse of mtr Theorem: If o-sgulr mtr, A, c be reduced to the ut mtr, I, b pre-multplcto wth strg of mtrces, the b pre-multplcto of the ut mtr, I, wth the sme mtr strg wll result the mtr verse, A - Cosder the mtr, A A T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 3

4 Computto of the determt d verse of mtr Fdg the verse: steps Ech step volves seres of le opertos Itermedte mtrces re obted usg pvot elemet elmto After step the termedte mtr A () A () 0 0 () () 2 22 () 2 () () 2 () T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 4

5 Computto of the determt d verse of mtr Elemets of mtr A () were computed s () j () j j j () j for provded tht 0. I geerl, t step k j, 2, A ( k ) ( k ), k ( k ) 2, k ( k ) 3, k ( k ) k, k ( k ), k ( k ), ( k ) 2, ( k ) 3, ( k ) k, ( k ), T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 5

6 Computto of the determt d verse of mtr Elemets of mtr A (k) were computed s ( k ) kj ( k ) j ( ( k) kj ( k) kk k) j ( k ) kj ( k) k for,, k j, Thus, strtg wth B=I d pplg the sme ops we get B () =A -. Mtrces B (k) re clculted usg: b b ( k ) kj ( k ) j b ( b ( k) kj ( k) kk k) j b ( k ) kj ( k) k for,, k j, T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 6

7 Computto of the determt d verse of mtr The determt of mtr A, A s: A = () 22 (2) 33 (-) becuse A A A () (2) ( ) () 22 A ( k ) A () A ( ) () 22 A () 22 (2) 33 ( k ) A T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 7

8 Computto of the determt d verse of mtr To mmze roudg errors: Before ech step, k, brg to posto k, b ter-chgg, the le cotg the elemet of bsolute mmum vlue colum k, selected from les umbered k, d bove B terchgg two les, the determt must be multpled b - T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 8

9 Computto of the determt d verse of mtr cost m = 0; tpe mtr= rr[..m,..m] of rel; procedure mtr_verse( :teger; out : mtr, out b:mtr; out det_:rel); vr, j, k, mpos: teger; u, m: rel; beg {Itlze mtr B=I} for := to do beg for j:= to do beg f (=j) the b[, j]=; else b[,j]=0; edf; det_:=; k:=; {k s the step computg mtrces A (k) d B (k) } T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 9

10 Computto of the determt d verse of mtr whle (k ) d (det_ 0) do beg {fd pvot le} m:= [k, k] ; mpos:=k; for :=k+ to do beg f [,k] > m the beg m:= [,k] ; mpos:=; edf; {terchge le k wth mpos mtrces A d B} f (k mpos) the beg for j:= to do beg u:=[k, j]; [k, j]:=[mpos, j]; [mpos,j]:=u; u:=b[k, j]; b[k,j]:=b[mpos,j]; b[mpos, j]:=u; det_:=-det_; edf; T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 0

11 Computto of the determt d verse of mtr f ([k, k]=0) the det_:=0; else beg det_:=det_*[k, k]; u:=[k, k]; for j:= to do beg [k, j]:=[k, j]/u; b[k, j]:=b[k, j]/u; for := to do beg u:=[, k]; f (k) the for j:= to do beg [, j]:=[, j]-[k, j]*u; b[, j]:=b[, j]-b[k, j]*u; edf; edf; k:=k+; ed. T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş

12 Appromto of fucto b poloml of degree m Problem: fucto s kow b ts vlues pots, (, ),, Appromto b poloml of degree m requred, P( ) p p p 2 such s the sum of the squres of the errors s mmum..e. B replcg P () t elds E 0 2 [ P( ) 2 )] 2 p m T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 2 m mml 2 ( p0 p p2 pm ) mml m

13 T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 3 Appromto of fucto b poloml of degree m The followg codtos result: Ad, fll the mtr equto 0 0;; 0; 0 p m E p E p E m m m m m m m m p p p

14 Notto: Appromto of fucto b poloml of degree m A = mtr of coeffcets P = mtr of ukow prmeters B = rght hd sde Rewrtte: AP=B Hece P=A - B As mtr verse lred hs lgorthm, we hve to: Geerte mtr A d vector B Multpl two mtrces T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 4

15 Appromto of fucto b poloml of degree m progrm lest_squres_method; cost m = ; tpe vector= rr[..m] of rel; mtr= rr[..m,..m] of rel; vr,, P, B: vector; {, hve elemets, P d B hve m+ elemets} A, A_v: mtr; {mtrces of order (m+)*(m+)}, m, m, : teger; det_a: rel; beg red, (([], []), =,); { s the umber of pots} red m; {m s the degree of the poloml} m:=m+; cll geerte_a(,,, m, A); cll geerte_b(,,, m, B); cll mtr_verse(m, A, A_v, det_a); f (det_a0) the beg cll mtr_product(m, m, A_v, B, P); wrte (P[], =,m); else wrte No-determto ; ed. edf; T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 5

16 Appromto of fucto b poloml of degree m procedure mtr_product(, m, p: teger,; A, B:mtr; out C:mtr); {compute product of mtrces A (of sze *m) d B (of sze m*p) to C} vr, j, k: teger; s: rel; beg for := to do beg for j:= to p do beg s:=0; for k:= to m do beg s:=s+a[, k] * B[k, j]; C[, j]:=s; T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 6

17 Appromto of fucto b poloml of degree m procedure geerte_a( : teger,; : vector; out A:mtr); vr, j, m: teger; s: vector; sum: rel; beg {geerte 2 2m,,, } s[]:=; m:=m-; for := to 2*m do beg sum:=0; for j:= to do beg sum:=sum+[j]^; s[+]:=sum; {geerte mtr A of sze m*m} for := to m do beg for j:= to m do beg A[, j]:=s[+j-]; T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 7

18 Appromto of fucto b poloml of degree m procedure geerte_b( : teger,;, : vector; out B:mtr); vr, j: teger; sum: rel; beg {clculte B[] } sum:=0; for := to do beg sum:=sum+[]; B[]:=sum; {compute B[2],,B[m+]} m:=m-; for := to m do beg sum:=0; for j:= to do beg sum:=sum+[j]^*[]; B[+]:=sum; T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 8

19 Numercl tegrto of the equto =f () Problem: tegrte =f () Soluto: for [ 0, r ] wth step h : fd vlues,, 2,,, whch ppromte the true vlues o pots, 2,,, where = 0 + *h of the fucto =f () whch cludes the tl pot ( 0, 0 ) Euler s method bsed o Tlor s ppromto: From whch the frst two terms re reted.e. ( 2 h ) ( ) h '( ) ''( ) 2! ( ) ( ) h '( ) h ( f, ) T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 9

20 Numercl tegrto of the equto =f () cost m=00; tpe vector=rr[0..m] of rel; fucto f(, : rel); beg f:=; {epresso of f} procedure Euler( :teger; 0, 0, f:rel; out, : vector); vr h: rel; : teger; beg h:=(f - 0) / ; [0]:=0; [0]:=0; for := to do beg []:=[-]+h; []=h*f([-], [-]); For cceptble precso, step h must be suffcetl smll T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 20

21 Numercl tegrto of the equto =f () A better ppromto results f frst three terms re tke. h [ 2 f (, ) f ( Thus we hve procedure Euler2( :teger; 0, 0, f:rel; out,: vector); vr h, f: rel; : teger; beg h:=(f - 0) / ; [0]:=0; [0]:=0; for := to do beg []:=[-]+h; f:=f([-], [-]); []=[-]+h/2*[f+f([], [-]+h*f);, h f (, T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 2 ))]

22 T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 22 Numercl tegrto of the equto =f () The predctor-corrector method combes the two methods s s computes the predcted vlue from the frst two terms of Tlor s seres d corrected vlue clculted from Tlor s seres from the frst three terms The lst formul s ppled repetedl tll predefed precso, (whch we wll cll eps), s met. At ech repetto, c + becomes the predcted vlue. A mmum umber of repettos wll be deoted b m_rep ), ( p f h ), ( ), ( 2 p c f f h

23 Numercl tegrto of the equto =f () procedure Euler3( m_rep: teger, 0, 0, f, eps: rel; out, : vector); vr, b_rep, j: teger; h, f, pred, corr, df: rel; go_o: Boole; beg h:=(f-0) / ; [0]:=0; [0]:=0; go_o:=true; whle ((<) d go_o) do beg []:=[-]+h; f:=f([-], [-]); pred:=[-]+h*f; b_rep:=0; repet b_rep:=b_rep+; corr:=[-]+h/2*[f+f([], pred)]; df:= corr-pred ; pred:=corr; utl (df <eps) or (b_rep > m_rep); f (b_rep > m_rep) the beg go_o:=flse; wrte No covergece wth the gve umber of tertos ; ed else beg []:=corr; :=+; ed edf T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 23

24 T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 24 Numercl tegrto of the equto =f (). Ruge-Kutt Aother method: proposed b Ruge d Kutt No detls. Just formuls for precso of the order h 4 : where k k k k ), ( f h k ) 2, 2 ( 2 3 k h f h k ) 2, 2 ( 2 k h f h k ), ( 3 4 k h f h k

25 Numercl tegrto of the equto =f (). Ruge-Kutt procedure Ruge_Kutt( : teger; 0, 0, f: rel; out, : vector); vr h, k, k2, k3, k4: rel; : teger; beg h:= (f - 0)/ ; [0]=0; [0]:=0; for :=0 to - do beg [+]:=[]+h; k:=h*f([], []); k2:=h*f([]+h/2, []+k/2); k3:=h*f([]+h/2, []+k2/2); k4:=h*f([]+h,[]+k3); [+]:=[]+/6*(k+2*k2+2*k3+k4); ed. T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 25

26 Smple progrms. Solvg ler equtos Problem: solve sstem of ler equtos wth vrbles () j j j b, () Method used: Guss,.e. use ler trsformtos to obt the row echelo (upper trgulr) form (2) j ' j where ' j j b' 0 for, 2, d j (2), T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 26

27 Smple progrms. Solvg ler equtos The equvlet sstem (2) s solved s: b' b' ' j ' ' j j for, (3) T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 27

28 Smple progrms. Solvg ler equtos The lgorthms s: beg *red put dt *brg equto sstem () to the row echelo (trgle) form (2) f sstem s comptble d determte the beg *solve sstem usg formuls (3) *dspl soluto ed edf ed. T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 28

29 Smple progrms. Solvg ler equtos Elemetr row opertos (lso kow s Guss opertos) re smple mpultos performed o ler sstem tht hve o effect o the sstem's soluto set. The elemetr row opertos re: swppg: equto s swpped wth other resclg: both sdes of equto re multpled b ozero costt pvotg: equto s replced b the sum of tself d multple of other T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 29

30 Smple progrms. Solvg ler equtos Brgg equto sstem () to the row echelo (trgle) form (2) c be doe s follows: Cosder les p, to be pvot les b brgg posto p (b performg terchges) the le cotg the mmum bsolute vlue elemet [p, p] Elmte ukow vrble [p] from equtos p, costt c d ddg tht le to le b multplg the pvot le p b [, p] [ p, p] for p, (4) T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 30

31 Smple progrms. Solvg ler equtos The we hve: *brg equto sstem () to the trgle form (2) s for p:= to - do beg *buld pvot le, p; for :=p+ to do beg c [,p] ; [p,p] ed ed. for j:=p to do beg [, j]:=[, j]+c*[p, j]; ed ed b[]:=b[]+c*b[p]; T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 3

32 Smple progrms. Solvg ler equtos The ostdrd pvot le buldg *buld pvot le, p s m:= [p, p] ; q:=p; for :=p+ to do beg f [, p] > m the beg m:=[, p]; q:=; ed edf ed {terchge of les p d q f tht s the cse} f p q the beg for j:=p to do beg t:=[p, j]; [p, j]:=[q, j]; [q, j]:=t; ed edf T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 32

33 Smple progrms. Solvg ler equtos The o-stdrd sttemet for solvg: *solve sstem usg formuls (3)s b[] [] ; [,] for :=- dowto do beg s:=0; for j:=+ to do beg s:=s+[, j]*[j]; ed ed [] b[]-s ; [,] T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 33

34 Smple progrms. Solvg ler equtos Modules of the lgorthm m progrm: progrm Guss; {solve ler equto sstem, =b, wth ukow vlues} cost m=0; tpe vector=rr[..m] of rel; mtr=rr[..m,..m] of rel; vr, j,, error: teger; : mtr; b, : vector; beg red, ((([, j], j=,), b[]), =,); cll row_echelo(,, b, error); {error=0 for uquel determed sstem, d otherwse} f error=0 the beg cll solve(,, b,); wrte([], =,); {soluto} ed else wrte udetermed or mpossble sstem: determt s zero. edf ed. T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 34

35 Smple progrms. Solvg ler equtos Modules of the lgorthm procedure row_echelo procedure row_echelo( :teger; out : mtr; out b: vector; out error: teger); vr p, q,, j: teger; c, m, t: rel; beg p:=; repet {buld pvot le, p} m:= [p, p] ; q:=p;{q cots mmum bsolute vlue colum p} for :=p+ to do beg f [, p] > m the beg m:=[, p]; q:=; ed edf ed f p q the beg {terchge of les p d q} for j:=p to do beg t:=[p, j]; [p, j]:=[q, j]; [q, j]:=t; ed t:=b[p]; b[p]:=b[q]; b[q]:=t; ed edf T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 35

36 Smple progrms. Solvg ler equtos Modules of the lgorthm procedure row_echelo (cot d) {trgle form} error:=0; f ([p, p]=0 the error:= else for :=p+ to do beg for j:=p to do beg [, j]:=[, j]+c*[p, j]; ed ed b[]:=b[]+c*b[p]; edf p:=p+; utl ((p=) or (error=) f ([, ]=0 the error:=; edf T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 36

37 Smple progrms. Solvg ler equtos Modules of the lgorthm procedure solve procedure solve( : teger; : mtr; b: vector; out : vector); vr, j: teger; s: rel; beg b[] [] ; [,] for :=- dowto do beg s:=0; for j:=+ to do beg s:=s+[, j]*[j]; ed ed ed. b[]-s [] ; [,] T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 37

38 Summr Smple progrms pseudocode Computto of the determt d verse of mtr Appromto of fucto b poloml of degree m usg lest squres method Numercl tegrto of the equto =f () wth tl codto ( 0 )= 0 Solvg ler equtos For ll smples: Problem defto Mth model Algorthm developmet T.U. Cluj-Npoc - Computer Progrmmg - lecture 2 - M. Joldoş 38