| 1 |
Imports System.Math
|
| 2 |
|
| 3 |
Module Outils_Math
|
| 4 |
|
| 5 |
|
| 6 |
|
| 7 |
|
| 8 |
#Region "calcul matriciel & vectoriel"
|
| 9 |
|
| 10 |
Public Function Addition(ByRef A(,) As Double, ByRef B(,) As Double) As Double(,)
|
| 11 |
'additionne deux matrices et le résultat est renvoyé dans la fonction
|
| 12 |
|
| 13 |
' vérification que l'on peut additionner, si toutes les matrices sont de 1 à nb, on devrait pas faire ça car ça bouffe du temps...
|
| 14 |
Dim ua1 As Double, ub1 As Double
|
| 15 |
Dim ua2 As Double, ub2 As Double
|
| 16 |
|
| 17 |
ua1 = UBound(A, 1)
|
| 18 |
ua2 = UBound(A, 2)
|
| 19 |
ub1 = UBound(B, 1)
|
| 20 |
ub2 = UBound(B, 2)
|
| 21 |
|
| 22 |
If (Not ua1 - ub1 = 0) Or (Not ua2 - ub2 = 0) Then
|
| 23 |
MsgBox("deux matrices ne peuvent pas s'additionner")
|
| 24 |
Return Nothing
|
| 25 |
End If
|
| 26 |
|
| 27 |
Dim matC(ua1, ua2) As Double
|
| 28 |
|
| 29 |
Dim i, j As Integer
|
| 30 |
For i = 0 To ub1
|
| 31 |
For j = 0 To ub2
|
| 32 |
matC(i, j) = A(i, j) + B(i, j)
|
| 33 |
Next j
|
| 34 |
Next i
|
| 35 |
|
| 36 |
Addition = matC
|
| 37 |
End Function
|
| 38 |
|
| 39 |
Public Function addition(ByRef a() As Double, ByRef b() As Double) As Double()
|
| 40 |
' addition de vecteurs
|
| 41 |
|
| 42 |
If Not UBound(a) = UBound(b) Then
|
| 43 |
MsgBox("Deux vecteurs ne peuvent s'additionner...")
|
| 44 |
Return Nothing
|
| 45 |
End If
|
| 46 |
|
| 47 |
Dim i As Integer
|
| 48 |
Dim c(UBound(a)) As Double
|
| 49 |
|
| 50 |
For i = 0 To UBound(a)
|
| 51 |
c(i) = a(i) + b(i)
|
| 52 |
Next
|
| 53 |
|
| 54 |
Return c
|
| 55 |
End Function
|
| 56 |
|
| 57 |
Public Function multiscal(ByRef A(,) As Double, ByRef c As Double) As Double(,)
|
| 58 |
' multiplie la matrice a par le scalaire c
|
| 59 |
Dim i As Integer
|
| 60 |
Dim j As Integer
|
| 61 |
|
| 62 |
Dim B(UBound(A, 1), UBound(A, 2)) As Double
|
| 63 |
|
| 64 |
For i = 0 To UBound(A, 1)
|
| 65 |
For j = 0 To UBound(A, 2)
|
| 66 |
B(i, j) = A(i, j) * c
|
| 67 |
Next j
|
| 68 |
Next i
|
| 69 |
|
| 70 |
multiscal = B
|
| 71 |
End Function
|
| 72 |
|
| 73 |
Public Function multiscal(ByRef A() As Double, ByRef c As Double) As Double()
|
| 74 |
'multiplie un vecteur par une certaine valeur
|
| 75 |
Dim i As Integer
|
| 76 |
Dim b(UBound(A)) As Double
|
| 77 |
For i = 0 To UBound(A)
|
| 78 |
b(i) = A(i) * c
|
| 79 |
Next
|
| 80 |
|
| 81 |
multiscal = b
|
| 82 |
End Function
|
| 83 |
|
| 84 |
Public Function reduit(ByRef A(,) As Double, Optional ByVal colonne As Integer = -1, Optional ByRef ligne As Integer = -1) As Double(,)
|
| 85 |
' fonction qui retire une colonne et /ou une ligne d'une matrice
|
| 86 |
Dim temp(UBound(A, 1) - 1, UBound(A, 2) - 1) As Double
|
| 87 |
Dim i As Integer, j As Integer, k As Integer, l As Integer
|
| 88 |
|
| 89 |
|
| 90 |
For l = 0 To UBound(A)
|
| 91 |
If Not (l = ligne) Then
|
| 92 |
j = 0
|
| 93 |
For k = 0 To UBound(A)
|
| 94 |
If Not (k = colonne) Then
|
| 95 |
temp(i, j) = A(l, k)
|
| 96 |
j += 1
|
| 97 |
End If
|
| 98 |
Next k
|
| 99 |
i += 1
|
| 100 |
End If
|
| 101 |
Next l
|
| 102 |
|
| 103 |
|
| 104 |
reduit = temp
|
| 105 |
End Function
|
| 106 |
|
| 107 |
Public Function reduit(ByRef A() As Double, Optional ByVal colonne As Integer = -1) As Double()
|
| 108 |
' fonction qui retire un élément d'un vecteur
|
| 109 |
Dim temp(UBound(A)) As Double
|
| 110 |
Dim i As Double, l As Double
|
| 111 |
|
| 112 |
While l < UBound(A)
|
| 113 |
If Not l = colonne Then
|
| 114 |
temp(i) = A(l)
|
| 115 |
l = l + 1
|
| 116 |
i = i + 1
|
| 117 |
Else
|
| 118 |
i = i + 1
|
| 119 |
End If
|
| 120 |
End While
|
| 121 |
|
| 122 |
Return temp
|
| 123 |
End Function
|
| 124 |
|
| 125 |
Public Function multiplication(ByVal A(,) As Double, ByVal B(,) As Double) As Double(,)
|
| 126 |
' le vecteur C est redimensionné correctement
|
| 127 |
' [A]*[B] = [C]
|
| 128 |
' si B est un vecteur alors on va à la partie en bas
|
| 129 |
|
| 130 |
Dim i, j, k, n, m As Integer
|
| 131 |
Dim c(,) As Double
|
| 132 |
|
| 133 |
m = UBound(B, 2)
|
| 134 |
n = UBound(A, 1)
|
| 135 |
|
| 136 |
If UBound(B, 1) <> UBound(A, 2) Then MsgBox(" Incompatibilité de matrices pour la multiplication") : Return Nothing
|
| 137 |
|
| 138 |
ReDim c(n, m)
|
| 139 |
|
| 140 |
For j = 0 To m
|
| 141 |
For i = 0 To n
|
| 142 |
For k = 0 To UBound(A, 2)
|
| 143 |
c(i, j) = c(i, j) + A(i, k) * B(k, j)
|
| 144 |
Next k
|
| 145 |
Next i
|
| 146 |
Next j
|
| 147 |
|
| 148 |
multiplication = c
|
| 149 |
End Function
|
| 150 |
|
| 151 |
Public Function multiplication(ByRef A(,) As Double, ByRef b() As Double) As Double()
|
| 152 |
' le vecteur C est redimensionné correctement
|
| 153 |
' [A]*{B} = {C}
|
| 154 |
' B est un vecteur ligne
|
| 155 |
|
| 156 |
Dim i, j, n, m As Integer
|
| 157 |
Dim c() As Double
|
| 158 |
|
| 159 |
m = UBound(b)
|
| 160 |
n = UBound(A, 1)
|
| 161 |
|
| 162 |
If UBound(A, 2) <> m Then MsgBox("Impossible de multiplier la matrice avec le vecteur, dimensions incompatibles!") : Return Nothing
|
| 163 |
|
| 164 |
' on multiplie une matrice et un vecteur
|
| 165 |
|
| 166 |
ReDim c(m)
|
| 167 |
|
| 168 |
For i = 0 To UBound(b)
|
| 169 |
For j = 0 To UBound(b)
|
| 170 |
c(i) = c(i) + A(i, j) * b(j)
|
| 171 |
Next j
|
| 172 |
Next i
|
| 173 |
Return c
|
| 174 |
End Function
|
| 175 |
|
| 176 |
Public Function multiplication(ByRef a() As Double, ByRef B(,) As Double) As Double()
|
| 177 |
' le vecteur C est redimensionné correctement
|
| 178 |
' {A}*[B] = {C}
|
| 179 |
' A est un vecteur colonne
|
| 180 |
|
| 181 |
Dim i, j, n, m As Integer
|
| 182 |
Dim c() As Double
|
| 183 |
|
| 184 |
m = UBound(a)
|
| 185 |
n = UBound(B, 2)
|
| 186 |
|
| 187 |
If m <> UBound(B, 1) Then MsgBox("Impossible de multiplier le veteur par la matrice.") : Return Nothing
|
| 188 |
|
| 189 |
ReDim c(m)
|
| 190 |
|
| 191 |
For i = 0 To UBound(B)
|
| 192 |
For j = 0 To UBound(B)
|
| 193 |
c(i) = c(i) + a(j) * B(j, i)
|
| 194 |
Next j
|
| 195 |
Next i
|
| 196 |
Return c
|
| 197 |
End Function
|
| 198 |
|
| 199 |
Public Function soustraction(ByRef A(,) As Double, ByRef B(,) As Double) As Double(,)
|
| 200 |
'C = A - B
|
| 201 |
Dim C(,) As Double
|
| 202 |
Dim m As Integer, n As Integer
|
| 203 |
m = UBound(A, 1)
|
| 204 |
n = UBound(A, 2)
|
| 205 |
|
| 206 |
If (m <> UBound(B, 1)) Or (n <> UBound(B, 2)) Then MsgBox("Impossible de soustraire les matrices, dimensions non compatibles") : Return Nothing
|
| 207 |
|
| 208 |
ReDim C(m, n)
|
| 209 |
|
| 210 |
Dim i, j As Integer
|
| 211 |
For i = 0 To m
|
| 212 |
For j = 0 To n
|
| 213 |
C(i, j) = A(i, j) - B(i, j)
|
| 214 |
Next j
|
| 215 |
Next i
|
| 216 |
Return C
|
| 217 |
End Function
|
| 218 |
|
| 219 |
Public Function transpose(ByRef A(,) As Double) As Double(,)
|
| 220 |
|
| 221 |
Dim c(,) As Double
|
| 222 |
Dim i, j As Integer
|
| 223 |
ReDim c(UBound(A, 2), UBound(A, 1))
|
| 224 |
|
| 225 |
For i = 0 To UBound(A, 1)
|
| 226 |
For j = 1 To UBound(A, 2)
|
| 227 |
c(j, i) = A(i, j)
|
| 228 |
Next j
|
| 229 |
Next i
|
| 230 |
|
| 231 |
Return c
|
| 232 |
End Function
|
| 233 |
|
| 234 |
Public Function gauss(ByRef A(,) As Double, ByRef C() As Double) As Double()
|
| 235 |
' [A]{X} = {C}
|
| 236 |
' les matrices doivent être déjà dimensionnées
|
| 237 |
Dim n As Integer
|
| 238 |
Dim i, j, k As Integer
|
| 239 |
Dim rapport As Double
|
| 240 |
|
| 241 |
n = UBound(A, 1)
|
| 242 |
Dim rep() As Double
|
| 243 |
ReDim rep(n)
|
| 244 |
|
| 245 |
|
| 246 |
For i = 0 To n ' de gauche à droite
|
| 247 |
pivoter(i, A, C)
|
| 248 |
|
| 249 |
'Call Form1.affiche(mat(), C())
|
| 250 |
'MsgBox i
|
| 251 |
|
| 252 |
For j = i + 1 To n ' les lignes
|
| 253 |
rapport = A(j, i) / A(i, i)
|
| 254 |
For k = 1 To n ' les colonnes
|
| 255 |
A(j, k) = A(j, k) - A(i, k) * rapport
|
| 256 |
Next k
|
| 257 |
C(j) = C(j) - C(i) * rapport
|
| 258 |
Next j
|
| 259 |
|
| 260 |
Next i
|
| 261 |
|
| 262 |
|
| 263 |
Dim stuff As Double
|
| 264 |
For i = n To 0 Step -1
|
| 265 |
stuff = 0
|
| 266 |
For j = i + 1 To n
|
| 267 |
stuff = stuff + A(i, j) * rep(j)
|
| 268 |
Next j
|
| 269 |
rep(i) = (C(i) - stuff) / A(i, i)
|
| 270 |
Next i
|
| 271 |
C = rep
|
| 272 |
gauss = rep
|
| 273 |
End Function
|
| 274 |
|
| 275 |
Private Sub pivoter(ByVal i As Integer, ByRef mat(,) As Double, ByRef C() As Double)
|
| 276 |
' sub qui pivote la ligne i de la matrice a ( et du vecteur C). Si et seulement si le pivotage est nécessaire.
|
| 277 |
Dim n As Integer
|
| 278 |
n = UBound(mat, 1)
|
| 279 |
Dim a_pivoter As Integer
|
| 280 |
|
| 281 |
Dim z As Integer
|
| 282 |
Dim temp As Double
|
| 283 |
Dim tempC As Double
|
| 284 |
|
| 285 |
For z = i To n
|
| 286 |
temp = mat(i, i)
|
| 287 |
|
| 288 |
If mat(z, i) > Math.Abs(temp) Then ' on a trouvé une ligneà pivoter
|
| 289 |
temp = mat(z, i)
|
| 290 |
a_pivoter = z
|
| 291 |
End If
|
| 292 |
Next z
|
| 293 |
'MsgBox a_pivoter
|
| 294 |
|
| 295 |
If a_pivoter <> 0 Then ' on pivote
|
| 296 |
' on pivote la ligne i et la ligne a_pivoter
|
| 297 |
tempC = C(i)
|
| 298 |
C(i) = C(a_pivoter)
|
| 299 |
C(a_pivoter) = tempC
|
| 300 |
|
| 301 |
For z = 0 To n
|
| 302 |
temp = mat(i, z)
|
| 303 |
mat(i, z) = mat(a_pivoter, z)
|
| 304 |
mat(a_pivoter, z) = temp
|
| 305 |
Next z
|
| 306 |
|
| 307 |
End If
|
| 308 |
' fin pivot
|
| 309 |
End Sub
|
| 310 |
|
| 311 |
Private Function pivoter(ByVal i As Integer, ByRef mat(,) As Double) As Integer
|
| 312 |
' sub qui pivote la ligne i de la matrice a ( et du vecteur C). Si et seulement si le pivotage est nécessaire.
|
| 313 |
Dim n As Integer
|
| 314 |
n = UBound(mat, 1)
|
| 315 |
Dim a_pivoter As Integer
|
| 316 |
pivoter = 1
|
| 317 |
|
| 318 |
Dim z As Integer
|
| 319 |
Dim temp As Double
|
| 320 |
|
| 321 |
For z = i To n
|
| 322 |
temp = mat(i, i)
|
| 323 |
|
| 324 |
If mat(z, i) > Math.Abs(temp) Then ' on a trouvé une ligne à pivoter
|
| 325 |
temp = mat(z, i)
|
| 326 |
a_pivoter = z
|
| 327 |
pivoter = -1
|
| 328 |
End If
|
| 329 |
Next z
|
| 330 |
|
| 331 |
If a_pivoter <> 0 Then ' on pivote
|
| 332 |
|
| 333 |
For z = 0 To n
|
| 334 |
temp = mat(i, z)
|
| 335 |
mat(i, z) = mat(a_pivoter, z)
|
| 336 |
mat(a_pivoter, z) = temp
|
| 337 |
Next z
|
| 338 |
|
| 339 |
End If
|
| 340 |
' fin pivot
|
| 341 |
End Function
|
| 342 |
|
| 343 |
Public Function norme(ByRef u() As Double) As Double
|
| 344 |
' calcul la norme (longueur) d'un vecteur à xDimensions
|
| 345 |
Dim i As Integer
|
| 346 |
Dim temp As Double
|
| 347 |
|
| 348 |
For i = 0 To UBound(u)
|
| 349 |
temp = temp + u(i) ^ 2
|
| 350 |
Next i
|
| 351 |
Return Math.Sqrt(temp)
|
| 352 |
End Function
|
| 353 |
|
| 354 |
Public Function Prod_scalaire(ByRef u() As Double, ByRef v() As Double) As Double
|
| 355 |
' fait le produit scalaire entre deux vecteurs
|
| 356 |
' en notation indicielle: w = uivi
|
| 357 |
Dim i As Integer
|
| 358 |
|
| 359 |
For i = 0 To UBound(u)
|
| 360 |
Prod_scalaire = Prod_scalaire + u(i) * v(i)
|
| 361 |
Next i
|
| 362 |
|
| 363 |
End Function
|
| 364 |
|
| 365 |
Public Function prod_vect3D(ByRef u() As Double, ByRef v() As Double) As Double()
|
| 366 |
' fait le produit vectoriel entre les vecteur 3D u et v
|
| 367 |
' [ i j k ]
|
| 368 |
' | u1 u2 u3 |
|
| 369 |
' [ v1 v2 v3 ]
|
| 370 |
'
|
| 371 |
' w = (u2*v3 - u3*v2) i - (u1*v3 - u3*v1) j + (u1*v2-u2*v1) k
|
| 372 |
Dim w(2) As Double
|
| 373 |
w(0) = u(2) * v(3) - u(3) * v(2)
|
| 374 |
w(1) = -(u(1) * v(3) - u(3) * v(1))
|
| 375 |
w(2) = u(1) * v(2) - u(2) * v(1)
|
| 376 |
|
| 377 |
prod_vect3D = w
|
| 378 |
|
| 379 |
End Function
|
| 380 |
|
| 381 |
Public Function unitaire(ByRef u() As Double) As Double()
|
| 382 |
' prend le vecteur u et son vecteur unitaire v
|
| 383 |
Dim longueur As Double
|
| 384 |
Dim i As Integer
|
| 385 |
Dim v() As Double
|
| 386 |
ReDim v(UBound(u))
|
| 387 |
|
| 388 |
Dim temp As Double
|
| 389 |
For i = 0 To UBound(u)
|
| 390 |
temp = temp + u(i) ^ 2
|
| 391 |
Next i
|
| 392 |
longueur = Math.Sqrt(temp)
|
| 393 |
For i = 0 To UBound(u)
|
| 394 |
v(i) = u(i) / longueur
|
| 395 |
Next i
|
| 396 |
|
| 397 |
unitaire = v
|
| 398 |
End Function
|
| 399 |
|
| 400 |
Public Function cosdir(ByRef u() As Double, ByRef v() As Double) As Double
|
| 401 |
' retourne le cosinus entre deux vecteurs, u et v
|
| 402 |
cosdir = Prod_scalaire(u, v) / norme(u) / norme(v)
|
| 403 |
End Function
|
| 404 |
|
| 405 |
Public Function projection(ByVal u() As Double, ByVal e() As Double) As Double
|
| 406 |
' projette le vecteur u sur e et retourne
|
| 407 |
' le résultat est la longueur sur le vecteur e
|
| 408 |
projection = norme(u) * cosdir(u, e)
|
| 409 |
End Function
|
| 410 |
|
| 411 |
|
| 412 |
|
| 413 |
Public Function det(ByRef A(,) As Double) As Double
|
| 414 |
' calcule le détrminant de la matrice A
|
| 415 |
Dim i As Long
|
| 416 |
Dim temp As Double
|
| 417 |
' contrôle d'erreur:
|
| 418 |
If UBound(A, 1) <> UBound(A, 2) Then
|
| 419 |
MsgBox("Impossible de calculer le déterminant, la matrice n'est pas carrée")
|
| 420 |
End If
|
| 421 |
|
| 422 |
Dim n As Integer
|
| 423 |
Dim j As Integer, k As Integer
|
| 424 |
Dim rapport As Double
|
| 425 |
|
| 426 |
n = UBound(A, 1)
|
| 427 |
Dim rep() As Double
|
| 428 |
ReDim rep(n)
|
| 429 |
|
| 430 |
temp = 1
|
| 431 |
For i = 0 To n ' de gauche à droite
|
| 432 |
temp *= pivoter(i, A)
|
| 433 |
|
| 434 |
For j = i + 1 To n ' les lignes
|
| 435 |
rapport = A(j, i) / A(i, i)
|
| 436 |
For k = 1 To n ' les colonnes
|
| 437 |
A(j, k) = A(j, k) - A(i, k) * rapport
|
| 438 |
Next k
|
| 439 |
Next j
|
| 440 |
Next i
|
| 441 |
|
| 442 |
' à partir d'ici on a une matrice triangulaire suppérieure. (mais je me casse pas la tête à mettre des 0...)
|
| 443 |
|
| 444 |
For i = 0 To UBound(A)
|
| 445 |
temp *= A(i, i)
|
| 446 |
Next
|
| 447 |
|
| 448 |
Return temp
|
| 449 |
End Function
|
| 450 |
|
| 451 |
|
| 452 |
Public Function det2(ByRef A(,) As Double) As Double
|
| 453 |
' calcul du déterminant à l'aide du cofacteur
|
| 454 |
|
| 455 |
Dim i As Long
|
| 456 |
Dim temp As Double
|
| 457 |
' contrôle d'erreur:
|
| 458 |
If UBound(A, 1) <> UBound(A, 2) Then
|
| 459 |
MsgBox("Impossible de calculer le déterminant, la matrice n'est pas carrée")
|
| 460 |
End If
|
| 461 |
|
| 462 |
If UBound(A) = 0 Then det2 = A(0, 0) : Exit Function
|
| 463 |
Dim B(,) As Double
|
| 464 |
ReDim B(UBound(A) - 1, UBound(A) - 1)
|
| 465 |
|
| 466 |
|
| 467 |
For i = 0 To UBound(A)
|
| 468 |
B = (reduit(A, i, 0))
|
| 469 |
temp += (-1) ^ i * det2(reduit(A, i, 0)) * A(0, i)
|
| 470 |
Next i
|
| 471 |
|
| 472 |
det2 = temp
|
| 473 |
|
| 474 |
End Function
|
| 475 |
|
| 476 |
|
| 477 |
'Function qui compare 2 vecteurs. retourne 0 si non alignés. 1 si dans le même sens, 2 si de sens opposé.
|
| 478 |
Public Function CompareSens(ByRef u() As Double, ByRef v() As Double, Optional ByVal Epsilon As Double = 0.000000001) As Integer
|
| 479 |
If ((u(0) - v(0)) < Epsilon) And ((u(1) - v(1)) < Epsilon) And ((u(2) - v(2)) < Epsilon) Then Return 1
|
| 480 |
If ((u(0) + v(0)) < Epsilon) And ((u(1) + v(1)) < Epsilon) And ((u(2) + v(2)) < Epsilon) Then Return 2
|
| 481 |
Return 0
|
| 482 |
End Function
|
| 483 |
|
| 484 |
#End Region
|
| 485 |
|
| 486 |
#Region "calcul intégral"
|
| 487 |
|
| 488 |
|
| 489 |
Function rectangles(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
|
| 490 |
' function qui calcule l'intégrale de f par la méthode des rectangles
|
| 491 |
Dim ecart As Double
|
| 492 |
Dim i As Integer
|
| 493 |
Dim nb As Long
|
| 494 |
Dim x As Double
|
| 495 |
Dim aire As Double
|
| 496 |
|
| 497 |
ecart = b - a
|
| 498 |
|
| 499 |
nb = CLng(ecart / dx)
|
| 500 |
dx = ecart / nb
|
| 501 |
|
| 502 |
If nb < 100 Then MsgBox("Attention, moins de 100 rectangles seront utilisés pour calculer l'intégrale")
|
| 503 |
|
| 504 |
x = a
|
| 505 |
For i = 1 To nb
|
| 506 |
aire += f(x + dx / 2) * dx
|
| 507 |
x += dx
|
| 508 |
Next
|
| 509 |
|
| 510 |
rectangles = aire
|
| 511 |
|
| 512 |
End Function
|
| 513 |
|
| 514 |
|
| 515 |
Function trapezes(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
|
| 516 |
' function qui calcule l'intégrale de f par la méthode des trapezes
|
| 517 |
Dim ecart As Double
|
| 518 |
Dim i As Integer
|
| 519 |
Dim nb As Long
|
| 520 |
Dim x As Double
|
| 521 |
Dim aire As Double
|
| 522 |
|
| 523 |
ecart = b - a
|
| 524 |
nb = CLng(ecart / dx)
|
| 525 |
dx = ecart / nb
|
| 526 |
|
| 527 |
If nb < 100 Then MsgBox("Attention, moins de 100 trapezes seront utilisés pour calculer l'intégrale")
|
| 528 |
|
| 529 |
x = a
|
| 530 |
For i = 1 To nb
|
| 531 |
aire += (f(x) + f(x + dx)) / 2 * dx
|
| 532 |
x += dx
|
| 533 |
Next
|
| 534 |
|
| 535 |
trapezes = aire
|
| 536 |
End Function
|
| 537 |
Function IntegrationGauss(ByRef n As Integer, Optional ByRef noFunction As Integer = 1) As Double
|
| 538 |
Dim i As Integer
|
| 539 |
Dim temp As Double
|
| 540 |
Dim t() As Double
|
| 541 |
Dim w() As Double
|
| 542 |
ReDim w(0)
|
| 543 |
ReDim t(0)
|
| 544 |
|
| 545 |
Select Case n
|
| 546 |
Case 1 ' un seul point d'intégration
|
| 547 |
ReDim t(0)
|
| 548 |
ReDim w(0)
|
| 549 |
|
| 550 |
t(0) = 0
|
| 551 |
w(0) = 2
|
| 552 |
|
| 553 |
Case 2 ' deux points d'intégration
|
| 554 |
' somme wi f(ri)
|
| 555 |
ReDim t(1)
|
| 556 |
ReDim w(1)
|
| 557 |
|
| 558 |
t(0) = -0.57735027
|
| 559 |
t(1) = 0.57735027
|
| 560 |
|
| 561 |
w(0) = 1
|
| 562 |
w(1) = 1
|
| 563 |
|
| 564 |
Case 3 ' trois points
|
| 565 |
ReDim t(2)
|
| 566 |
ReDim w(2)
|
| 567 |
|
| 568 |
t(0) = -0.77459667
|
| 569 |
t(1) = 0
|
| 570 |
t(2) = 0.77459667
|
| 571 |
|
| 572 |
w(0) = 5 / 9
|
| 573 |
w(1) = 8 / 9
|
| 574 |
w(2) = 5 / 9
|
| 575 |
|
| 576 |
Case 4
|
| 577 |
|
| 578 |
ReDim t(3)
|
| 579 |
ReDim w(3)
|
| 580 |
|
| 581 |
t(0) = -0.86113631
|
| 582 |
t(1) = -0.33998104
|
| 583 |
t(2) = 0.33998104
|
| 584 |
t(3) = 0.86113631
|
| 585 |
|
| 586 |
w(0) = 0.34785485
|
| 587 |
w(1) = 0.65214515
|
| 588 |
w(2) = 0.65214515
|
| 589 |
w(3) = 0.34785485
|
| 590 |
|
| 591 |
|
| 592 |
Case 5
|
| 593 |
ReDim t(4)
|
| 594 |
ReDim w(4)
|
| 595 |
|
| 596 |
t(0) = -0.90617975
|
| 597 |
t(1) = -0.53846931
|
| 598 |
t(2) = 0
|
| 599 |
t(3) = 0.53846931
|
| 600 |
t(4) = 0.90617975
|
| 601 |
|
| 602 |
w(0) = 0.23692689
|
| 603 |
w(1) = 0.47862867
|
| 604 |
w(2) = 0.56888889
|
| 605 |
w(3) = 0.47862867
|
| 606 |
w(4) = 0.23692689
|
| 607 |
|
| 608 |
Case Else
|
| 609 |
MsgBox("Problème dans l'intégration de Gauss, nb de points d'intégration non reconnu!?!")
|
| 610 |
End Select
|
| 611 |
|
| 612 |
|
| 613 |
For i = 0 To n - 1
|
| 614 |
temp = temp + w(i) * Choose(noFunction, f1(t(i)), f2(t(i)), f3(t(i)), f4(t(i)), f5(t(i)))
|
| 615 |
Next
|
| 616 |
|
| 617 |
IntegrationGauss = temp
|
| 618 |
End Function
|
| 619 |
|
| 620 |
|
| 621 |
Function simpson(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
|
| 622 |
|
| 623 |
|
| 624 |
End Function
|
| 625 |
|
| 626 |
#End Region
|
| 627 |
|
| 628 |
|
| 629 |
|
| 630 |
|
| 631 |
Public Function f(ByVal x As Double) As Double
|
| 632 |
f = x ^ 2
|
| 633 |
End Function
|
| 634 |
Public Function f1(ByVal r As Double) As Double
|
| 635 |
f1 = -0.04 * (-3 + r) * r ^ 2
|
| 636 |
End Function
|
| 637 |
Public Function f2(ByVal r As Double) As Double
|
| 638 |
f2 = -0.04 * (-3 + r) * r ^ 2
|
| 639 |
End Function
|
| 640 |
Public Function f3(ByVal r As Double) As Double
|
| 641 |
f3 = -0.04 * (-3 + r) * r ^ 2
|
| 642 |
End Function
|
| 643 |
Public Function f4(ByVal r As Double) As Double
|
| 644 |
f4 = -0.04 * (-3 + r) * r ^ 2
|
| 645 |
End Function
|
| 646 |
Public Function f5(ByVal r As Double) As Double
|
| 647 |
f5 = -0.04 * (-3 + r) * r ^ 2
|
| 648 |
End Function
|
| 649 |
|
| 650 |
|
| 651 |
' Secant
|
| 652 |
Public Function Sec(ByVal X As Double) As Double
|
| 653 |
Sec = 1 / Math.Cos(X)
|
| 654 |
End Function
|
| 655 |
|
| 656 |
' Cosecant
|
| 657 |
Public Function CoSec(ByVal X As Double) As Double
|
| 658 |
CoSec = 1 / Math.Sin(X)
|
| 659 |
End Function
|
| 660 |
|
| 661 |
' Cotangent
|
| 662 |
Public Function CoTan(ByVal X As Double) As Double
|
| 663 |
CoTan = 1 / Math.Tan(X)
|
| 664 |
End Function
|
| 665 |
|
| 666 |
' Inverse Sine
|
| 667 |
Public Function ArcSin(ByVal X As Double) As Double
|
| 668 |
ArcSin = Math.Atan(X / Math.Sqrt(-X * X + 1))
|
| 669 |
End Function
|
| 670 |
|
| 671 |
' Inverse Cosine
|
| 672 |
Public Function ArcCos(ByVal X As Double) As Double
|
| 673 |
ArcCos = Math.Atan(-X / Math.Sqrt(-X * X + 1)) + 2 * Math.Atan(1)
|
| 674 |
End Function
|
| 675 |
|
| 676 |
' Inverse Secant
|
| 677 |
Public Function ArcSec(ByVal X As Double) As Double
|
| 678 |
ArcSec = Math.Atan(X / Math.Sqrt(X * X - 1)) + Math.Sign(X - 1) * (2 * Math.Atan(1))
|
| 679 |
End Function
|
| 680 |
|
| 681 |
' Inverse Cosecant
|
| 682 |
Public Function ArcCoSec(ByVal X As Double) As Double
|
| 683 |
ArcCoSec = Math.Atan(X / Math.Sqrt(X * X - 1)) + (Math.Sign(X) - 1) * (2 * Math.Atan(1))
|
| 684 |
End Function
|
| 685 |
|
| 686 |
' Inverse Cotangent
|
| 687 |
Public Function ArcCoTan(ByVal X As Double) As Double
|
| 688 |
ArcCoTan = Math.Atan(X) + 2 * Math.Atan(1)
|
| 689 |
End Function
|
| 690 |
|
| 691 |
|
| 692 |
' Hyperbolic Secant
|
| 693 |
Public Function HSec(ByVal X As Double) As Double
|
| 694 |
HSec = 2 / (Math.Exp(X) + Math.Exp(-X))
|
| 695 |
End Function
|
| 696 |
|
| 697 |
' Hyperbolic Cosecant
|
| 698 |
Public Function HCoSec(ByVal X As Double) As Double
|
| 699 |
HCoSec = 2 / (Math.Exp(X) - Math.Exp(-X))
|
| 700 |
End Function
|
| 701 |
|
| 702 |
' Hyperbolic Cotangent
|
| 703 |
Public Function HCotan(ByVal X As Double) As Double
|
| 704 |
HCotan = (Math.Exp(X) + Math.Exp(-X)) / (Math.Exp(X) - Math.Exp(-X))
|
| 705 |
End Function
|
| 706 |
|
| 707 |
' Inverse Hyperbolic Sine
|
| 708 |
Public Function HArcSin(ByVal X As Double) As Double
|
| 709 |
HArcSin = Math.Log(X + Math.Sqrt(X * X + 1))
|
| 710 |
End Function
|
| 711 |
|
| 712 |
' Inverse Hyperbolic Cosine
|
| 713 |
Public Function HArcCos(ByVal X As Double) As Double
|
| 714 |
HArcCos = Math.Log(X + Math.Sqrt(X * X - 1))
|
| 715 |
End Function
|
| 716 |
|
| 717 |
' Inverse Hyperbolic Tangent
|
| 718 |
Function HArcTan(ByVal X As Double) As Double
|
| 719 |
HArcTan = Math.Log((1 + X) / (1 - X)) / 2
|
| 720 |
End Function
|
| 721 |
|
| 722 |
' Inverse Hyperbolic Secant
|
| 723 |
Public Function HArcSec(ByVal X As Double) As Double
|
| 724 |
HArcSec = Math.Log((Math.Sqrt(-X * X + 1) + 1) / X)
|
| 725 |
End Function
|
| 726 |
|
| 727 |
' Inverse Hyperbolic Cosecant
|
| 728 |
Public Function HArcCoSec(ByVal X As Double) As Double
|
| 729 |
HArcCoSec = Math.Log((Math.Sign(X) * Math.Sqrt(X * X + 1) + 1) / X)
|
| 730 |
End Function
|
| 731 |
|
| 732 |
' Inverse Hyperbolic Cotangent
|
| 733 |
Public Function HArcCoTan(ByVal X As Double) As Double
|
| 734 |
HArcCoTan = Math.Log((X + 1) / (X - 1)) / 2
|
| 735 |
End Function
|
| 736 |
|
| 737 |
' Logarithm to base N
|
| 738 |
Public Function LogN(ByVal X As Double, ByVal n As Double) As Double
|
| 739 |
LogN = Math.Log(X) / Math.Log(n)
|
| 740 |
End Function
|
| 741 |
|
| 742 |
' function qui effectue la rotation d'un point Pt par la matrice Mat
|
| 743 |
Public Function Rotation2D(ByRef Mat(,) As Double, ByVal Pt() As Double) As Double()
|
| 744 |
Dim temp(1) As Double
|
| 745 |
|
| 746 |
temp(0) = Mat(0, 0) * Pt(0) + Mat(0, 1) * Pt(1)
|
| 747 |
temp(1) = Mat(0, 1) * Pt(0) + Mat(1, 1) * Pt(1)
|
| 748 |
|
| 749 |
Return temp
|
| 750 |
|
| 751 |
End Function
|
| 752 |
|
| 753 |
' function qui effectue la rotation d'un point Pt par le vecteur UNITAIRE u
|
| 754 |
Public Function Rotation2D(ByRef u() As Double, ByVal Pt() As Double) As Double()
|
| 755 |
Dim temp(1) As Double
|
| 756 |
Dim longueur As Double = Math.Sqrt(u(0) * u(0) + u(1) * u(1))
|
| 757 |
u(0) /= longueur
|
| 758 |
u(1) /= longueur
|
| 759 |
|
| 760 |
temp(0) = u(0) * Pt(0) - u(1) * Pt(1)
|
| 761 |
temp(1) = u(1) * Pt(0) + u(0) * Pt(1)
|
| 762 |
|
| 763 |
|
| 764 |
|
| 765 |
Return temp
|
| 766 |
|
| 767 |
End Function
|
| 768 |
|
| 769 |
Public Sub Rotation2D(ByRef u() As Double, ByRef Ptx As Double, ByRef Pty As Double)
|
| 770 |
Dim temp(1) As Double
|
| 771 |
Dim longueur As Double = Math.Sqrt(u(0) * u(0) + u(1) * u(1))
|
| 772 |
u(0) /= longueur
|
| 773 |
u(1) /= longueur
|
| 774 |
|
| 775 |
temp(0) = u(0) * Ptx - u(1) * Pty
|
| 776 |
temp(1) = u(1) * Ptx + u(0) * Pty
|
| 777 |
|
| 778 |
Ptx = temp(0)
|
| 779 |
Pty = temp(1)
|
| 780 |
|
| 781 |
End Sub
|
| 782 |
|
| 783 |
|
| 784 |
''' <summary>
|
| 785 |
''' Retourne l'angle entre 2 vecteurs
|
| 786 |
''' </summary>
|
| 787 |
''' <param name="u">Le premier vecteur</param>
|
| 788 |
''' <param name="v">Le second vecteur</param>
|
| 789 |
''' <returns>Un angle en radian</returns>
|
| 790 |
''' <remarks>Retourne Pi si les 2 vecteurs sont d'orientation identique mais de direction opposés</remarks>
|
| 791 |
Public Function Angle2Vecteurs(ByRef u() As Double, ByRef v() As Double) As Double
|
| 792 |
Dim temp As Double
|
| 793 |
temp = Outils_Math.Prod_scalaire(u, v) / norme(u) / norme(v)
|
| 794 |
If Math.Abs(temp - 1) < 0.000000000001 Or Math.Abs(temp + 1) < 0.000000000001 Then
|
| 795 |
Return Pi
|
| 796 |
Else
|
| 797 |
Return Math.Acos(temp)
|
| 798 |
End If
|
| 799 |
End Function
|
| 800 |
|
| 801 |
|
| 802 |
''' <summary>
|
| 803 |
''' Compare 2 vecteur de dimension 3
|
| 804 |
''' </summary>
|
| 805 |
''' <param name="u">Premier Vecteur</param>
|
| 806 |
''' <param name="v">Second vecteur</param>
|
| 807 |
''' <returns>0 si sans rapport, 1 si identiques, 2 si sens inverse, 3 si même sens mais norme différente, 4 si norme et sens différent</returns>
|
| 808 |
''' <remarks>Non optimisé pour les vecteurs unitaires</remarks>
|
| 809 |
Public Function ComparerVecteurs3D(ByRef u() As Double, ByRef v() As Double) As Byte
|
| 810 |
|
| 811 |
If u.GetUpperBound(0) <> 2 AndAlso v.GetUpperBound(0) <> 2 Then Return 0
|
| 812 |
If Math.Abs(u(0) - v(0)) < 0.0005 AndAlso Math.Abs(u(1) - v(1)) < 0.0005 AndAlso Math.Abs(u(1) - v(1)) < 0.0005 Then Return 1
|
| 813 |
If Math.Abs(u(0) + v(0)) < 0.0005 AndAlso Math.Abs(u(1) + v(1)) < 0.0005 AndAlso Math.Abs(u(1) + v(1)) < 0.0005 Then Return 2
|
| 814 |
|
| 815 |
Dim u2() As Double = unitaire(u)
|
| 816 |
Dim v2() As Double = unitaire(v)
|
| 817 |
|
| 818 |
If Math.Abs(u2(0) - v2(0)) < 0.0005 AndAlso Math.Abs(u2(1) - v2(1)) < 0.0005 AndAlso Math.Abs(u2(1) - v2(1)) < 0.0005 Then Return 3
|
| 819 |
If Math.Abs(u2(0) + v2(0)) < 0.0005 AndAlso Math.Abs(u2(1) + v2(1)) < 0.0005 AndAlso Math.Abs(u2(1) + v2(1)) < 0.0005 Then Return 4
|
| 820 |
|
| 821 |
Return 0
|
| 822 |
|
| 823 |
End Function
|
| 824 |
|
| 825 |
''' <summary>
|
| 826 |
''' Fonction qui inverse le sens d'un vecteur
|
| 827 |
''' </summary>
|
| 828 |
''' <param name="u">Le vecteur</param>
|
| 829 |
''' <returns>Un vecteur de direction inverse</returns>
|
| 830 |
''' <remarks></remarks>
|
| 831 |
Public Function InverserVecteur(ByRef u() As Double) As Double()
|
| 832 |
Dim v() As Double
|
| 833 |
ReDim v(u.GetUpperBound(0))
|
| 834 |
|
| 835 |
For i As Integer = 0 To u.GetUpperBound(0)
|
| 836 |
v(i) = -u(i)
|
| 837 |
Next
|
| 838 |
Return v
|
| 839 |
End Function
|
| 840 |
|
| 841 |
|
| 842 |
|
| 843 |
End Module
|
