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DrSplinter/prubeh_fce_1D.ipynb

Last active November 18, 2021 20:26
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matika - machy
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prubeh_fce_nD.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from sympy import *\n",
"from IPython.display import display\n",
"\n",
"def latex_double_dollar(cl):\n",
" f = cl._repr_latex_\n",
" cl._repr_latex_ = lambda *args, **kwargs : f\"${f(*args,**kwargs)}$\"\n",
"\n",
"if 'times' not in globals():\n",
" for c in [Eq, Set, Matrix]:\n",
" latex_double_dollar(c)\n",
"times = True"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Zadani, pro funkci $$f$$ najdete Hessovu matici a gradient"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\displaystyle f{\\left(x_{1},x_{2},x_{3} \\right)} = x_{1}^{2} + x_{1} x_{2} - 6 x_{1} + 1.5 x_{2}^{2} - 4 x_{2} + 3 x_{3}^{2} + x_{3}$$"
],
"text/plain": [
"Eq(f(x1, x2, x3), x1**2 + x1*x2 - 6*x1 + 1.5*x2**2 - 4*x2 + 3*x3**2 + x3)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"syms = symbols('x1:4')\n",
"f = Function('f')\n",
"x1, x2, x3 = syms\n",
"fun = x1**2 + 1.5*x2**2 + 3*x3**2 + x1*x2 + x3 - 4*x2 - 6*x1\n",
"Eq(f(x1,x2,x3), fun)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Krok 1 - Vypocet parcialnich derivaci (gradient)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\displaystyle \\left[\\begin{matrix}2 x_{1} + x_{2} - 6\\\\x_{1} + 3.0 x_{2} - 4\\\\6 x_{3} + 1\\end{matrix}\\right]$$"
],
"text/plain": [
"Matrix([\n",
"[ 2*x1 + x2 - 6],\n",
"[x1 + 3.0*x2 - 4],\n",
"[ 6*x3 + 1]])"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"grad = Matrix(derive_by_array(fun, syms))\n",
"grad\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Krok 2 - Vypocet druhych parcialnich derivaci (Hessova matice)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\displaystyle \\left[\\begin{matrix}2 & 1 & 0\\\\1 & 3.0 & 0\\\\0 & 0 & 6\\end{matrix}\\right]$$"
],
"text/plain": [
"Matrix([\n",
"[2, 1, 0],\n",
"[1, 3.0, 0],\n",
"[0, 0, 6]])"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hess = Matrix(derive_by_array(derive_by_array(fun, syms), syms))\n",
"hess"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Krok 3 - Pozitivne definitni"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\displaystyle \\left|{\\left[\\begin{matrix}2\\end{matrix}\\right]}\\right| = 2$$"
],
"text/plain": [
"Eq(Determinant(Matrix([[2]])), 2)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\displaystyle \\left|{\\left[\\begin{matrix}2 & 1\\\\1 & 3.0\\end{matrix}\\right]}\\right| = 5.0$$"
],
"text/plain": [
"Eq(Determinant(Matrix([\n",
"[2, 1],\n",
"[1, 3.0]])), 5.0)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$\\displaystyle \\left|{\\left[\\begin{matrix}2 & 1 & 0\\\\1 & 3.0 & 0\\\\0 & 0 & 6\\end{matrix}\\right]}\\right| = 30.0$$"
],
"text/plain": [
"Eq(Determinant(Matrix([\n",
"[2, 1, 0],\n",
"[1, 3.0, 0],\n",
"[0, 0, 6]])), 30.0)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"for i in range(len(syms)):\n",
" s = hess[:i+1,:i+1]\n",
" display(Eq(Determinant(s), s.det(),evaluate=False))\n",
"hess.is_positive_definite"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Krok 4 - stacionarni body"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\displaystyle \\left\\{\\left( 2.8, \\ 0.4, \\ -0.166666666666667\\right)\\right\\}$$"
],
"text/plain": [
"{(2.8, 0.4, -0.166666666666667)}"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"linsolve(grad,syms)"
]
}
],
"metadata": {
"interpreter": {
"hash": "aee8b7b246df8f9039afb4144a1f6fd8d2ca17a180786b69acc140d282b71a49"
},
"kernelspec": {
"display_name": "Python 3.9.7 64-bit",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.7"
},
"orig_nbformat": 4
},
"nbformat": 4,
"nbformat_minor": 2
}
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