︠c8e6f40f-4d53-478c-ba96-c59dc9ecf297asi︠ %auto typeset_mode(True, display=False) ︡04eb2799-1d84-41df-b25e-059788a1b40a︡{"done":true} ︠7298fef0-2a10-429b-880b-a3f73ddb5ffei︠ %md ## Descente de gradient Dans les pages 108 à 110 du livre de Stewart on présente une méthode de minimisation, connue sous le nom de "descente de gradient". En gros, on part d'un point puis on fait des pas dans la direction oposée au gradient. La longueur du pas est celle qui permet de minimiser la valeur de la fonction objectif (si elle existe...). Ci après, une implémentation très naïve de la chose, en SAGEMath. L'implémentation ci bas fait appel à la fonction `find_local_minimum(...)` qui retourne une liste de valeurs. ### L'exemple utilisé est celui de la page 109 - 110. ︡3e19b314-4bc2-4790-b654-a2c6384cdfeb︡{"done":true,"md":"## Descente de gradient\n Dans les pages 108 à 110 du livre de Stewart on présente une méthode de minimisation, connue sous le nom de \"descente de gradient\". En gros, on part d'un point puis on fait des pas dans la direction oposée au gradient. La longueur du pas est celle qui permet de minimiser la valeur de la fonction objectif (si elle existe...). Ci après, une implémentation très naïve de la chose, en SAGEMath.\n\nL'implémentation ci bas fait appel à la fonction `find_local_minimum(...)` qui retourne une liste de valeurs.\n\n### L'exemple utilisé est celui de la page 109 - 110."} ︠20f64c19-ff78-4a1c-a245-933b030e5907s︠ var('x,y') f(x,y) = x^4+y^2 -2*x^2 *y+2*y+x ︡3a3c5a94-90e3-4253-ae50-e5134b516f83︡{"stdout":"(x, y)\n"}︡{"done":true}︡ ︠fd2eb1f9-db1c-4882-b7e9-3501b5b0ee2c︠ ︡1628d382-1285-4b9b-ab40-49b303cc60e2︡ ︠7f2762ae-8012-484c-9eb7-f13334a2754cs︠ var('t') def myDescente(f, prec, x0,y0):# prec = précision, x0, y0 sont les coordonnées du point de départ. Df = f.gradient() print "Point d'évalution :", (x0,y0) print "Valeur de la fonction objectif:", f(x0,y0).n(digits=6), "Norme du gradient = ", norm(Df(x0,y0)).n(digits=6) if norm(Df(x0,y0)) < prec : print "Valeur minimale trouvée. Arrêt." else: F(t) = f(x0-t*Df(x0,y0)[0],y0-t*Df(x0,y0)[1]) t0 = F.find_local_minimum(0,2)[1] print "On va ailleurs. Longueur du pas (en fraction du gradient) : ", t0 myDescente(f,prec,x0-t0*Df(x0,y0)[0], y0-t0*Df(x0,y0)[1]) ︡dc9d569a-fc4c-4d12-a37b-66dc7ac261e5︡{"html":"
$t$
"}︡{"done":true}︡ ︠e9b7f446-6a3c-406a-adc2-c931590ec020s︠ myDescente(f,0.01, 0,0) ︡32c1d1a3-a046-4918-9d22-fce085cbe25b︡{"stdout":"Point d'évalution : (0, 0)\nValeur de la fonction objectif: 0.000000 Norme du gradient = 2.23607\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.380408919629\nPoint d'évalution : (-0.38040891962887613, -0.7608178392577523)\nValeur de la fonction objectif: -1.08206 Norme du gradient = 0.422488\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.345102226762\nPoint d'évalution : (-0.2499999964962533, -0.8260222923108163)\nValeur de la fonction objectif: -1.11257 Norme du gradient = 0.249272\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.301870420298\nPoint d'évalution : (-0.28365182325811644, -0.8933259383163823)\nValeur de la fonction objectif: -1.12205 Norme du gradient = 0.117240\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.320913060169\nPoint d'évalution : (-0.2499999930467315, -0.910151862449617)\nValeur de la fonction objectif: -1.12425 Norme du gradient = 0.0611523\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.302753560077\nPoint d'évalution : (-0.25827974829437783, -0.9267113565644658)\nValeur de la fonction objectif: -1.12482 Norme du gradient = 0.0294276\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.314569974107\nPoint d'évalution : (-0.24999999981318474, -0.9308512327243619)\nValeur de la fonction objectif: -1.12496 Norme du gradient = 0.0148671\nOn va ailleurs. Longueur du pas (en fraction du gradient) : 0.30296365379\nPoint d'évalution : (-0.25201433489340147, -0.93487990243502)\nValeur de la fonction objectif: -1.12499 Norme du gradient = 0.00719511\nValeur minimale trouvée. Arrêt.\n"}︡{"done":true}︡ ︠c253d29e-1daf-4636-b3ab-f624a1c6e810i︠ %md ### Voyons maintenant l'exercice n.31, section 3.1 (celui du devoir). D'abord avec le point de départ $(1/4, 1/4)$, le critère d'arêt est que la norme du gradient soit inférieure à $10^{-6}$. ︡84209d43-dc48-4267-a7c9-1370e4bfaeef︡{"done":true,"md":"### Voyons maintenant l'exercice n.31, section 3.1 (celui du devoir). \n\nD'abord avec le point de départ $(1/4, 1/4)$, le critère d'arêt est que la norme du gradient soit inférieure à $10^{-6}$."} ︠ebea2904-05c8-4726-bd0f-cb752fe7b34fs︠ ︡97b9da3c-f36b-49d5-95f0-d0209457e0ad︡{"done":true}︡ ︠793df690-8c7f-4f10-839b-b40c214a7ef7s︠ g(x,y) = x^2+ y^2 - (x^2+y^2)^(3/2) myDescente(g, 10^-6, 0.25,0.25) ︡9bd3852e-a1b9-4ca2-a3d4-2b08fc81fdd6︡{"stdout":"Point d'évalution : (0.250000000000000, 0.250000000000000)\nValeur de la fonction objectif: 0.0808058 Norme du gradient = "}︡{"stdout":"0.332107\nOn va ailleurs. Longueur du pas (en fraction du gradient) : "}︡{"stdout":" 1.06457745783\nPoint d'évalution : (-1.59089003082258e-9, -1.59089003082258e-9)\nValeur de la fonction objectif: 5.06186e-18 Norme du gradient = 4.49972e-9\nValeur minimale trouvée. Arrêt.\n"}︡{"done":true}︡ ︠b64626ec-ed2a-4ccd-afe7-fe767f754414i︠ %md ### À partir du point (1,1) Les choses sont très différentes : la fonction $h(t) = f((1,1) + t \nabla f(1,1))$ n'a pas de minimums locaux, la descente ne s'arête jamais. L'algorithme ne converge pas (on peut essayer de l'exécuter, voir ce que ça donne) ︡91ad3d8d-3116-49da-83f8-e1a66eb2bdf3︡{"done":true,"md":"### À partir du point (1,1) \nLes choses sont très différentes : la fonction $h(t) = f((1,1) + t \\nabla f(1,1))$ n'a pas de minimums locaux, la descente ne s'arête jamais. L'algorithme ne converge pas (on peut essayer de l'exécuter, voir ce que ça donne)"} ︠8920306d-93dc-4abe-9f8c-fa085646011es︠ #myDescente(g, 10^-6, 1,1) Dg = g.gradient() h(t) = g(1 - t* Dg(1,1)[0], 1 - t* Dg(1,1)[1]) plot(h,0,1) ︡bac6d150-64a0-42d2-8842-6b4323fa056f︡{"file":{"filename":"/home/user/.sage/temp/project-4aacee0b-64bd-4d37-99a1-d87b7a2c4cd6/117/tmp_MEg0Qa.svg","show":true,"text":null,"uuid":"7d23963f-19fb-47aa-a2e4-5487639ba1c1"},"once":false}︡{"done":true}︡ ︠1ad1fa14-c17c-4de9-98ef-8bba83775203︠ ︡bccd87ab-2edb-4572-9266-5feecb4299d2︡ ︠b2b25854-b96a-4d4a-a0da-4c8543827044i︠ %md ### Voyons la surface, on comprendra ce qui est arrivé. ︡ce29ffef-5ee1-4fc9-9c02-a4493445c781︡{"done":true,"md":"### Voyons la surface, on comprendra ce qui est arrivé."} ︠08d800ac-f893-4d89-b3a4-f8dc1ac0d915︠ ︡bc47e52b-e0a0-4b14-8319-69375b42d487︡ ︠9d42bdf5-cf46-4020-b5c6-86eb54cfa06as︠ Surf = plot3d(g,(x,-1.1,1.1), (y,-1.1,1.1), mesh=0.7) show(Surf) ︡d586c1e4-3e2b-4c02-82f7-dfc740aa7282︡{"file":{"filename":"e3967c03-aeaa-4ee9-82ad-868784d50a49.sage3d","uuid":"e3967c03-aeaa-4ee9-82ad-868784d50a49"}}︡{"done":true}︡ ︠b98c5b10-1b33-4868-8ec2-f1de6c787d5ci︠ %md ### Multiplicateurs de Lagrange (ex. 38, section 3.3) On doit maximiser $f(x,y) = 2x+3y$ avec la contrainte $g(x,y) = \sqrt{x} + \sqrt{y} = 5$. Voyons quelques courbes de niveau de la fonction objectif, ainsi que la courbe de contrainte ︡5b0c5083-ccab-434f-8252-a861cb834f24︡{"done":true,"md":"### Multiplicateurs de Lagrange (ex. 38, section 3.3)\nOn doit maximiser $f(x,y) = 2x+3y$ avec la contrainte $g(x,y) = \\sqrt{x} + \\sqrt{y} = 5$. Voyons quelques courbes de niveau de la fonction objectif, ainsi que la courbe de contrainte"} ︠c73317cb-14a8-47d6-a6c8-2a428ae41591s︠ var('x,y') C = contour_plot(2*x+3*y, (x,0,15), (y,0,15), cmap = "Blues", colorbar=True, fill=True) Cg = plot((5-sqrt(x))^2, (x,0,15),ymin = 0, ymax = 15, color = "red" ) show(Cg+C) ︡77d3e781-d807-480a-b7f3-70b2cc3d8794︡{"html":"
($x$, $y$)
"}︡{"file":{"filename":"/home/user/.sage/temp/project-4aacee0b-64bd-4d37-99a1-d87b7a2c4cd6/117/tmp_syh7e2.svg","show":true,"text":null,"uuid":"03f78204-82a2-488a-a3ca-133ee75532f3"},"once":false}︡{"done":true}︡ ︠f3a24448-a7eb-4f67-80f2-dadc2a7d132fi︠ %md ### Si on commence suffisament près du "creux", la méethode de descente du gradient donne des bons résultats. Ce n'Est pas le cas si on commence loin. ︡b2576cb4-dfaa-465c-a6ef-63ac05dcc7b0︡ ︠de9872da-b133-4f98-a981-18250f9cda06i︠ %md ## Exercice n.25, page 246 Il s'agit de donner des approximations de la solution de $y'(x) = 6x^2 -3x^2 y$, avec $y(0)= 3$. On utilise la méthode d'Euler. ︡d31248f0-fda0-455c-a169-d7f2fc0ae4e1︡{"done":true,"md":"## Exercice n.25, page 246\nIl s'agit de donner des approximations de la solution de $y'(x) = 6x^2 -3x^2 y$, avec $y(0)= 3$. On utilise la méthode d'Euler."} ︠58d3fd5c-b60a-4dc4-8325-a6b8dbe8325ds︠ V=[3] ︡a807a398-f8af-4b7f-a8f7-5afe3459b718︡{"done":true}︡ ︠ef6bbb14-5fae-4cda-82f6-488c7da1908bs︠ var('x,y') def F(x,y) : return 6*x^2 - 3 * x^2 * y #Définition de la fonction h = 0.0001 #Le pas N = 10000+1 #Le nombre de points x0 = 0 #L'abscisse du point de départ y0 = 3 #L'ordonnée du point de départ X = [x0 + j*h for j in range(N)] # Les abscisses des points Y = [y0] for j in range(N) : Y = Y + [Y[j-1] + h * F(X[j],Y[j]) ] #La boucle pour calculer les ordonnées Points = [(X[j], Y[j]) for j in range(N)] Points[-1] V = Y + [Points[-1][1]] # Juste pour avoir la liste des valeurs estimées de y(1). Pas très élégant comme code ︡2db4f09f-ec67-48d5-b45f-d93360149593︡{"html":"
($x$, $y$)
"}︡{"html":"
($1.00000000000000$, $2.60653065982095$)
"}︡{"done":true} ︠63d5c824-e008-404e-9465-8c199a3e57ee︠ ︡2c4d650a-f63a-4db8-b80e-76c93b0f46e9︡ ︠271bb2ee-9178-462b-afed-7046a1f0e45ds︠ f(x)= 2+e^(-x^3) ︡1c5ae083-1811-464c-9cfb-2acd4d82ec94︡{"done":true} ︠67f67fa3-4a0e-4b14-b90b-97ff420ee54ci︠ %md On évalue maintenant l'erreur commise entre les approximations et la "vraie valeur" ︡2db9465b-57de-4c30-a72b-1d5538b39d01︡{"done":true,"md":"On évalue maintenant l'erreur commise entre les approximations et la \"vraie valeur\""} ︠b15fa35c-7a7f-40fe-873f-d68123e168efs︠ [(v-f(1)).n(digits=10) for v in V] ︡f531a8d0-cc46-478b-85f2-71753afa2186︡{"html":"
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$0.6250767951$, $0.6250680832$, $0.6250593642$, $0.6250506381$, $0.6250419048$, $0.6250331644$, $0.6250244169$, $0.6250156622$, $0.6250069003$, $0.6249981314$, $0.6249893552$, $0.6249805719$, $0.6249717815$, $0.6249629839$, $0.6249541791$, $0.6249453671$, $0.6249365480$, $0.6249277218$, $0.6249188883$, $0.6249100477$, $0.6249011999$, $0.6248923449$, $0.6248834827$, $0.6248746133$, $0.6248657368$, $0.6248568530$, $0.6248479621$, $0.6248390640$, $0.6248301586$, $0.6248212460$, $0.6248123263$, $0.6248033993$, $0.6247944652$, $0.6247855238$, $0.6247765751$, $0.6247676193$, $0.6247586562$, $0.6247496860$, $0.6247407084$, $0.6247317237$, $0.6247227317$, $0.6247137325$, $0.6247047260$, $0.6246957123$, $0.6246866914$, $0.6246776632$, $0.6246686277$, $0.6246595850$, $0.6246505350$, $0.6246414778$, $0.6246324133$, $0.6246233416$, $0.6246142626$, $0.6246051763$, $0.6245960827$, $0.6245869819$, $0.6245778737$, $0.6245687583$, $0.6245596356$, $0.6245505057$, $0.6245413684$, $0.6245322238$, $0.6245230720$, $0.6245139128$, $0.6245047463$, $0.6244955726$, $0.6244863915$, $0.6244772031$, $0.6244680074$, $0.6244588044$, $0.6244495941$, $0.6244403764$, $0.6244311515$, $0.6244219192$, $0.6244126796$, $0.6244034326$, $0.6243941783$, $0.6243849166$, $0.6243756477$, $0.6243663714$, $0.6243570877$, $0.6243477[...]","stderr":"\nWARNING: Output: 160059 truncated by MAX_HTML_SIZE to 40000. Type 'smc?' to learn how to raise the output limit."}︡{"done":true} ︠7aabdcb1-3185-446d-9aa9-9973e5c430cai︠ %md ### La question n. 26, section 6.3 L'équation différentielle s'écrit $y e^y y' = x\sqrt{x^2 +1}$, il s'agit d'une équation à variables séparables. On peut intégrar sans trop de mal et trouver la solution exacte. Faisons le avec SAGE ︡8b35d422-1c6a-482f-b6e3-39c46d262a1b︡{"done":true,"md":"### La question n. 26, section 6.3\n\nL'équation différentielle s'écrit $y e^y y' = x\\sqrt{x^2 +1}$, il s'agit d'une équation à variables séparables. On peut intégrar sans trop de mal et trouver la solution exacte. Faisons le avec SAGE"} ︠af0e44dd-cdf0-45f1-ba33-34e0cd014b1es︠ var('x,y') integrate(y*e^y,y) integrate(x*sqrt(x^2 +1),x) ︡99dd1608-15f2-4292-8000-28957c849490︡{"html":"
($x$, $y$)
"}︡{"html":"
${\\left(y - 1\\right)} e^{y}$
"}︡{"html":"
$\\frac{1}{3} \\, {\\left(x^{2} + 1\\right)}^{\\frac{3}{2}}$
"}︡{"done":true}︡ ︠5f964a7f-8503-4759-8fa1-b6bf26dca3f8i︠ %md Les courbes solution sont donc les courbes $(y-1)e^y = \frac{1}{3}(x^2 + 1 )^{3/2} + C$ dessinons-en queleques unes ︡022f9a37-5c45-4890-b700-0b88ba85f1ea︡{"done":true,"md":"Les courbes solution sont donc les courbes $(y-1)e^y = \\frac{1}{3}(x^2 + 1 )^{3/2} + C$ dessinons-en queleques unes"} ︠e46d1eb0-cfdd-4641-a000-1e5c3f0b58c5︠ ︡b39dc3d6-8ca0-434a-901f-f93541d348af︡ ︠9e8873ac-2e7a-4faa-b4ef-52b89f6e1ed7s︠ contour_plot((y-1)*e^y - (1/3)*(x^2+1)^(3/2), (x,-5,5), (y,-5,5), cmap="autumn", fill = False, contours = range(-10,10) + [10 * (k+1) for k in range(5)], labels = True, colorbar = True) ︡4d8a379f-2b99-4cb3-9a38-5cb90cb8a0c1︡{"file":{"filename":"/home/user/.sage/temp/project-4aacee0b-64bd-4d37-99a1-d87b7a2c4cd6/117/tmp__ysSDp.svg","show":true,"text":null,"uuid":"e2f813bb-e23e-4d43-b439-c26e04e75eb9"},"once":false}︡{"done":true}︡ ︠1026cb2b-ee1c-45ac-b4ad-9cd46b306b32i︠ %md ### Exercice n. 31, section 6.3. Il s'agit de trouver les courbes orthogonales à la famille de courbes $y = \frac{k}{x}$. Les courbes de la famille cherchée ont pour équation $y^2 = x^2 +c$. Voyons quelques courbes de la famille originale, et de la famille des trajectoires orthogonales. ︡5ec43070-a25b-4317-922b-cedbb29254f5︡{"done":true,"md":"### Exercice n. 31, section 6.3.\nIl s'agit de trouver les courbes orthogonales à la famille de courbes $y = \\frac{k}{x}$. Les courbes de la famille cherchée ont pour équation $y^2 = x^2 +c$. Voyons quelques courbes de la famille originale, et de la famille des trajectoires orthogonales."} ︠1ba04636-a9d0-49fc-bf6d-dacd7ab28ba3s︠ var('x,y') ListeCourbes = [implicit_plot( x*y - c , (x,-3,3), (y,-3,3), color="red") for c in range(-5,5)] FamilleOrthogonale = [implicit_plot(y^2 - x^2 - c, (x,-3,3), (y,-3,3),color="blue") for c in range(-5,5) ] show(sum(ListeCourbes) + sum(FamilleOrthogonale)) ︡d6eee941-50c1-4455-92f6-949e55a51b23︡{"html":"
($x$, $y$)
"}︡{"file":{"filename":"/home/user/.sage/temp/project-4aacee0b-64bd-4d37-99a1-d87b7a2c4cd6/117/tmp_EbGAPv.svg","show":true,"text":null,"uuid":"7a0c0d2f-9214-49b4-8c33-a0c246fabe92"},"once":false}︡{"done":true}︡ ︠8b318c06-e693-4f82-9bbf-3e467d2a3039︠