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"source": [
"# `scipy`\n",
"A large collection of functions related to scientific computing and data analysis. Makes heavy use of `numpy` and provides a Python interface to many F77 libraries. These slides give a non-exhaustive overview of the functionality."
]
},
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"source": [
"%matplotlib inline\n",
"%config InlineBackend.figure_formats = ['svg']\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Special functions\n",
"A large number of special functions is available in `scipy.special`. All of them accept scalars and numpy arrays as arguments."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"from scipy import special"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Some Examples\n",
"#### $\\Gamma$ function"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8.8417619937397008e+30"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"special.gamma(30)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Recall that for integer values: $n! = \\Gamma(n+1)$."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
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"0.0"
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"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"special.factorial(29) - special.gamma(30)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Factorials are actually approximated using $\\Gamma$. We can force exact integer evaluation to see the error introduced by floating-point arithmetic"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1125899906842624.0"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"special.factorial(29, exact=True) - special.gamma(30)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is of the order of $10^{-16}$ as expected."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Special Polynomials\n",
"These return a polynomial object with the coefficients.\n",
"#### Legendre polynomials"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 3\n",
"2.5 x - 1.5 x\n"
]
}
],
"source": [
"p = special.legendre(3)\n",
"print(p)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"It can be evaluated like a function."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"data": {
"text/plain": [
"-0.43749999999999994"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p(0.5)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Derivatives can be computed directly from the coefficients."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 2\n",
"7.5 x - 1.5\n"
]
}
],
"source": [
"print(p.deriv())"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Chebychev polynomials\n",
"`chebyt` are the polynomials of first kind ($T_n$), `chebyu` of second kind ($U_n$)."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 5 4 3 2\n",
"16 x - 1.776e-15 x - 20 x + 2.665e-15 x + 5 x - 3.062e-16\n"
]
}
],
"source": [
"print(special.chebyt(5))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Let's check the recurrence relation.\n",
"$$T_{n+1} = 2x T_n - T_{n-1}$$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 5 3 2\n",
"16 x - 20 x + 2.22e-15 x + 5 x - 1.837e-16\n"
]
}
],
"source": [
"print(special.chebyt(4) * 2 * np.poly1d([1,0]) - special.chebyt(3))"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 5 4 3 2\n",
"16 x - 1.776e-15 x - 20 x + 2.665e-15 x + 5 x - 3.062e-16\n"
]
}
],
"source": [
"print(special.chebyt(5))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"## Spherical harmonics\n",
"The implementation in `sph_harm` does *not* use the Condon–Shortley phase ($(-1)^m$ for $m>0$) common in quantum mechanics."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"slideshow": {
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},
"metadata": {},
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}
],
"source": [
"phi, theta = np.meshgrid(np.linspace(-np.pi,np.pi),np.linspace(0,np.pi))\n",
"gr = plt.GridSpec(3,5,wspace=0)\n",
"for l in range(3):\n",
" for m in range(-l,l+1):\n",
" ax = plt.subplot(gr[l,m+2], projection='mollweide')\n",
" ax.set_title('l={}, m={}'.format(l,m))\n",
" ax.pcolormesh(phi, 0.5 * np.pi - theta,\n",
" np.real(special.sph_harm(m,l,phi,theta)),\n",
" rasterized=True)\n",
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]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Constants\n",
"Many physical constants are available in the `constants` module. The data are taken from the CODATA 2014 recommendations. Values are given in SI units. More advanced handling of units attached to values is provided by the `astropy.units` package."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from scipy import constants"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.67408e-11"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants.gravitational_constant # also constants.G"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.022140857e+23"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants.Avogadro # also constants.N_A"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Additional information is available in the dictionary `physical_constants`. Each value also has information on its unit and uncertainty."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(8.3144598, 'J mol^-1 K^-1', 4.8e-06)"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants.physical_constants['molar gas constant']"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(6.67408e-11, 'm^3 kg^-1 s^-2', 3.1e-15)"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants.physical_constants['Newtonian constant of gravitation']"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Unit conversion\n",
"A simple function to convert temperatures is provided. See `astropy.units` for a complete unit conversion package."
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"32.0"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants.convert_temperature(0.,'C', 'F')"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"text/plain": [
"1.8"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"constants.convert_temperature(1,'K','R')\n",
"# Kelvin to Rankine"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Interpolation\n",
"The `interpolate` module provides several polynomial and spline interpolators as well as interpolators for multidimensional data. It also provides an interface to several FITPACK routines."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from scipy import interpolate"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"# generate some data\n",
"x = np.linspace(0, 2 * np.pi, 101) # fine grid for plotting\n",
"f = lambda x: np.sin(x) + 0.5 * np.cos(3.5 * x)\n",
"y = f(x)\n",
"xd = x[::10]; yd = y[::10] # coarse data for fitting"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
},
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"### Simple 1D interpolation"
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" interp = interpolate.interp1d(xd, yd, kind=k)\n",
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"source": [
"### Generic Interface\n",
"`interpolate.UnivariateSpline` provides a generic object-oriented interface to splines. It supports up to fifth order and can apply smoothing to data. Data points can be given individual weights (e.g., error estimates)."
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"quint = interpolate.UnivariateSpline(xd, yd, k=5, s=0)\n",
"quints = interpolate.UnivariateSpline(xd, yd, k=5, s=0.1)\n",
"plt.plot(x, quint(x), label='quintic')\n",
"plt.plot(x, quints(x), label='quintic smoothed')\n",
"plt.plot(x, y, ls=':', label='original')\n",
"plt.legend()"
]
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"slide_type": "subslide"
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"source": [
"### Monotonicty\n",
"PCHIP provides a cubic splines which preserves monotonicity in data and does not overshoot on discontinuous data."
]
},
{
"cell_type": "code",
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"#### Discontinuous data\n",
"In the last example PCHIP performed worse than the cubic spline because the sampled data were monotonous even though the original function was not. Its strength shows when applied to discontinous functions."
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"source": [
"### Derivatives\n",
"Splines naturally provide derivates. The argument to `derivate` specifies the order of the derivate. -1 computes the antiderivative."
]
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"cubicd = cubic.derivative(); pchipd = pchip.derivative()\n",
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"gd = lambda x: np.where(x<np.pi, np.cos(x), -np.sin(x))\n",
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{
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"source": [
"#### Smooth example"
]
},
{
"cell_type": "code",
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"metadata": {
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"cubicd = interpolate.CubicSpline(xd, f(xd)).derivative()\n",
"pchipd = interpolate.PchipInterpolator(xd, f(xd)).derivative()\n",
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"source": [
"### Extrapolation\n",
"By default the interpolators raise an exception when applied outside the range of input data. We can set a fixed value for this range or allow extrapolation."
]
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" fill_value='extrapolate')\n",
"extrax = np.linspace(-0.5 * np.pi, 3 * np.pi)\n",
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"source": [
"### Polynomial interpolation\n",
"$n$ data points can be interpolated with a single polynomial function of degree $n-1$."
]
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"polycheb = interpolate.BarycentricInterpolator(xc,yc)\n",
"plt.plot(x,poly(x), label='equidistant')\n",
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"source": [
"### Multidimensional data\n",
"The interpolate module offers bivariate splines with variable degree in both dimensions. This only supports Cartesian geometry."
]
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"# generate data\n",
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"# coarse grid\n",
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"Z = h(X, Y)\n",
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6UYe2O/ECzMaNg8GctaQzcU/WigVfOrRbZiuLLG5MVv6Hi/1GHZ4FW7EZ2agVhzazTwcyWAYCEyNaYLOvDGc1bWf5coxPlesmMSH41uHHMUdFuUxcKDea14TgOszIuIGBhL6KfLUAXEpgThsTABvWdOXgdQdZxLFFT6J+9HO0iLLYsf3BJO53PbJdAcbxurepSbaFazu0iF4dnq9MLdfHMiPWBBP3KpE8laz5QHKCkZgi4iShKEFeUe4hW1q/HiyPN2PSlsc4sHy5Aows0R8bZjSvZztY4Jm3kEvExZRGLdIdM7MRX5712972/7R+t2XLzlrJVK0481lBkShHWTxVRzShMrZYhoEgWryTxf5xsVJPZ759cJXDKzQNZ/6RrMWdY5CSmclniHCd2l3GI383b83Am2bNjRZ4QPsadLTBqwCfVuVDMBsrKeS+0ox6buzyeh6blWoOS1kcYlE9i5FlOPkmNC7W/BCJBOyAKHKy0uRcjx8xghGlO/kd9/7ZI8cz2CqQ6kIkLoEZOXofDl5/QL1W3/wuTJz+ThbbR0rhAhXI4QFgAarGxDEKXtsYrsW6aUS/H4xULWWsIZFNQS5doi3BJhBdChG5P/TMz15ffYGgRIvw9ZY1ZqbVH/OJJB8ty7dj8RXNWVq4+MpkoX5VpS7gpVuUWNCCEYwo31Hk2/4T6zjmuHYeu1iA6npbOCRUllI65Lxy4ZGMEPuNY19KbnpK5J5ERVlsH8lUAm2+ivKLHAaTSbSpS7IWM7HTJkjkkkv4/f1GIllzp7RSxjomzldbxw4le0KS66nR9rE9E735+Ve5RCk8tacyPdqrw++B7UfjmCneOdlsmJXnPGD8cUZ70jjghZkhioTVl+9kcW2Vd4gchaksPgL9RNdcxZDInDFz53fdSWTrW7ToUggSyT0BxI/aJroABnXUDOWv4Wv1likL3CQMTfhc/wJ7J5JZLezygq+Sj7g+e0L7icMCFCXI+9e4RDaFUQIcbIeS7YdW4OF8TCKXyExddPPnqRFFdtHox4gSt1Dc5fItkW/7T2wtfkIeUeF+P3XMsS5P2aANDj9II1FNrqJE4hzF3mppzXFs8VGjxRgS2mRRfgFDBYNdLNyE+JEVidyAwWBD1ugnSmb5zmCkWgrXnFktqJoMFgtrKdFz1O+g+lXIXYIzrvGrzaihLMascPJZY9RbTippYvEMPPX2IZkbazBTvHNGlPWZKz0BCzhGGasmAqb6XWpiRZevWX2hlMLVTv1GaVzvg9k9TGXeYihUq35XKbm5gLTa1wNoFNEiyqL8StlbEpkAWn09JbK3CjB/qq0VsKEPdtpiuewDqaBFrcEkSHjaODXI5fc5QqKNZ/hmWdF1veWSXzaLe0GatWbNLOrHes9TUSXqALwOz8NvT9N8bpHhB7A94Zk/y/5QSnLtRYkJe0xZvCXybT9qXbjuBOu13IW1wIF/XbYyHXuURv+L7dO8xd5SOV5Kbg7VD9LBsQdn/kUWQxqd1TrkwZDBoF220uHQq5IPA2Mi2Dgd82CaEjlLucR6mTkWByEiMpnxedFnRgwz9CUoiXPMs2V5j02m1tPhDzYDMu0tV/LpklLprDaNg826oMOfqypikPfkjc0olS33JCAp2ol7/6TflVvlYbvBgm0rikygXh9eIkecDBvPtEFZs0Ab/C6MHEuE2HiLHDM0dX/rHmDujwHAGP0xgUrXe6L3Caam4ASpZhs2Gyur3gHAvB+HqLkkcS3CPlYiiXARI0RLJZ5wHwHrTfMZ+K0aGgMZlxSmy+Onu55zPv9nZOkbPa81MNZeUa++/Fsi3/aj1jG52sq4JOyHjY49n9uNc216XHgE1+fChUhwxnXIYoPyG2A2bjDkg8WCj00WDxIYTNZ7/JRfGew6BmWR8yRPDr5bbpCu2TbNCSGMpanAiC6baGrxvmDjq9+pqhZLWtwjg3un0+Ke4qQSjN6ZUyrzeSULLinM8dJ85tn+BIdfeNTC1UjAvu1tP2hdCPUS0v1lrYNtZ4wyhJT9OfDtjmL+aMuSSuCPvcy4/gPHHrPzmOfiR6/MdX0pX+UZyGDccrPV2y2lic5+mOXWUWYaqYTSFiZ78cEEiptvDGudm60F9cl9MHD4PX5oi9H8Pvq6FQL3WhtRgwEGIiorP2KqSdRKMPckr7enuskrA15g2MjvXZHIuhAKjkvmcFLHqBNGOTrXdej95mXui8tFHR17iBwL2Hqtoc/cFgAM81w7momIH4+UzdaJu28skbvKrZ3nXC5vcmBbUX7IIkFEqYOi+tDfc2Nr5Cfn87PP6X/0BNgDSn6aReTEnYlHgsrKfTS/pHQ1ZnVfiIi6WFl7rIGT78Olq8gkwVY2dgWwZbDIb4l8239jvYaU96yF/ZiOkAOb5a8jWQtLrZBgyuIjpX3DWsBm1JGpcnigOPkHx577AySy5y8P2rgDyvrvIZOvKJGTOCohhMxnzeIM2pNhSgMKFcksr82yfow6k00/X4N7lI47txVbEFWJbEI0gJXitmHw5UtDMeWzpZrVRxfpeYOdkEhMrq4PdsDIUWux3YVvjlmfxTuoUP5OclmixWyvMcL8oOJ33YAKAeR/QxZbh8gR9iOXXF2I0Qe7kTgWCnCYCSRdIUKEDdSPVlao29o+xgNgs2bhaxnXiDJ8VoguMYiVBjmx47IERCe/uwssf75hIZ9/+uxEBEnXt73tB+2SSHfyobIC5I8IAqAXGc3XZT+Wz2yG48hLIq/vF3Ds8dfZ84NK3Txz5f8SpsJoOl2shlLoy1S1Hj/xtR85SqQ7rZCT2swsB7VDIi0ruZj1tRBCM71uRSK1OPxxPaNev0nel5yoy8FsLHlPpRnh+D0+A6L6jPZn+Ltnvo7zKJK3qpyL/dabUkfxjZmyqJFmBdDrFzPl63ebHKBEJqgohmcYZLRIZM8aeu4NwAmAKHIJUhh7kNchodiPXHoB1W0keZmZxk011QI2P1rnMz/Hp+571EpXVIgA84FDLHhcNTfrI7oGqNaYpn8O3Du2vKcokdszWMfmAipMG6gO/jXf4KLKI78l8m3/jXVkLawBq2yG/dWAZODXwdGHanaUSwGmWlk8aEMUeSeRWE+PK/9GJQRnnzvzRSIfmfuCyLEw4o1EMrdkLaGtSrAfu4mu3Jnl+Rkkd0sVhl/b1LNENiYrsmDRZkG5zFuEty7eBs8tIlE+s49QDQSzH5Ku0N/9pPHAaB/Ahv/H7e9qBPJ68uVCt4subQjcEHh8viMAwn3v8RoBrr4DkDZfS0JGG51kkVlKoaEfs9YoXWBTHGbEdMflo7H0iC5tP9cYOGyrMvZwzVGgKHy8Xy/39OYZ+OvnZ2jQzy8YIKqg8v/nfTkT0tve9q9av6sUkK2/shY4eSU/to49UfKLYx9tQD1EP/hmFjqGN8w3k2P3qggiWsNAECFiu8jiK7OhfRVR5nVkDb2xxndZyKLE+CSzgEQC3VzH+XVZoZk49eznZDaXx9UNcgn3/agy5Xl98ZxPbHaDo14OupHCn7aMFqlyKtzsI6gw0hJOecFiwUPJDcpf3fG3Fl/i8eV84f/ug9XjJUFg4A9e7zPK0haTPFdDiSyyKGewvdwj74N7+sN2J537Kb3tbT9iVx6sRoy7IV3+ie30e+d4nozvmOzOkEEwityZZTu+jDfeMNVXBYduBsNDK7rUbG9buzBDuQ5IJEMZz7coaHMvjod8EVD5X/nDR4tyWftrVEm0SeR3P/x0Qit89WP+7DMTbN944ylbGJ/zxUPZAfjd9/1DCwmEJEX4Ykq3iMj3gX1xzcfPkEw3/InV7P3ra+vvnz3ct0S+7UftjxkMTf4Bvf6JfYtY/uQEvvGB3xl/PL8PHf/fVFB8lzG/xVbXd37vE//I/qkiof2/AKZQ1fsTFvP/fncSf/qBXx1SkqjfB5nZ/D2w/uA8iOhb1/bdj/onplb//hN7S+TbftT+mMHWorGHfvrniM/NO3Hq6e0J1KK88jlZH//6JVBHr1pUCiO+f8tObGZYfHhDPUcW/OKaj5+huW7Ynxgq0n4W+Wz/7IP7+rB8kx4ucIHqz88Y78vdAml3FG8KQPnOd+8AikpTmJSBAIzXZ5bc2CwRJaYebr/2cAz2mc2Xi0QQrYFvnx3+ewBuH/Tbw3GtsP0Z+N8/JQTHwo4JxI5j6i2Rb/tRKxKpWyyy//+nLIBvVLl5X5+B6FrNOWdTF+mAhUhihjTOVYyPnTEuuI71RR12Jz+ZLfvaDWvlzG6Dz6z9sxy/zhtmgeMG30UW8fM2yin3yPt+1vlHO2EE+/pd5LNLZexPA3JpZIUufWXjE/3i2iB1y+H0uxadv77ZlHKaITgKBWz4xfBlZWq/H8tb2x+8UpTcfGVH/8ryc/bq1rICz/U6AQBxVlGVvQOo1HIgHR1fyxlJ+zbPp/tensvxeX3xnI+rXJ9x9JbIt/2odSJgDdodfnvpw/9jRLmIxfs5+tCZP7Z35x9+nWXvH5/ZfDPNqyxbCVJTnP1rlwMTIZ9aZiblJx+7zIoek647ewVrFVnMnFhhtmCwkZ+zy6m35zxeZ5k7qXa8X18FVHtw9UKO8Qw3NruRQr3Bjp9jr516BJWW/v1LXi8kqBX2KsQLeblolL8LnBU8sLGBZZvKas6w0lprIEfgi/msH5gIqwQ0LuCSwgtmehyvLNGg6ga29R06n9GO70WJ3NcVi6U34drQXcD23H9seR9PK07vzyDeBs+tHPtyxZ6OchDWqDGfe/a/JfJtP2pXHmwhbpdH3M4t2neySODkU74eezpb3cak0Dn8UC1+nbBUpMH2eMPI2vXCgNU8cE7hhEWx2GURC/MkJZSVdD5XN1EM7hnrKhakK3I8/p6rRGJg4Z+5S+C6tlFYzWvvbYK0wsYONiaZLx0wYHc2dOAn3LuXe3rdoo3w1uv5PA36tTj22/EHXOxS6Zjqy0dIavMbabu+4hcTgipf1/3YO1k0iMYnkfnaorAckQGobMCWeLtE+o3HtbsZAZaVo5G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},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.subplot(121); plt.axis('equal'); plt.title('fine')\n",
"plt.pcolormesh(X, Y, Z, rasterized=True)\n",
"plt.subplot(122); plt.axis('equal'); plt.title('spline')\n",
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]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Unstructrured data\n",
"`LinearNDInterpolator` takes data sampled at arbitrary points and interpolates on a triangulated grid."
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# random sampling points\n",
"Xp = np.random.uniform(-1, 1,50)\n",
"Yp = np.random.uniform(-1, 1,50)"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"ndinterp = interpolate.LinearNDInterpolator(np.array([Xp, Yp]).T, h(Xp, Yp), fill_value=0.)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
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"## Root finding\n",
"The `optimize` module provides routines for root finding and fitting."
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" fjac: array([[-1.]])\n",
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"source": [
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{
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{
"data": {
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" fjac: array([[-1.]])\n",
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" message: 'The solution converged.'\n",
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{
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{
"data": {
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" fjac: array([[-1.]])\n",
" fun: array([ 2.22044605e-16])\n",
" message: 'The solution converged.'\n",
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},
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"source": [
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]
},
{
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"execution_count": 49,
"metadata": {},
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{
"data": {
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" nfev: 22\n",
" qtf: array([-0.24502514])\n",
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},
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"source": [
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{
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"We can use other algorithms to avoid getting stuck in local minima. Some of these make use of backtracking."
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"data": {
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" fun: array([ -1.01421047e-08])\n",
" message: 'A solution was found at the specified tolerance.'\n",
" nit: 10\n",
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"### More advanced options\n",
"Some functions might take additional (fixed) parameters apart from `x`. We can pass them as a tuple."
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" fjac: array([[-1.]])\n",
" fun: array([ 0.])\n",
" message: 'The solution converged.'\n",
" nfev: 6\n",
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"source": [
"#### Callbacks\n",
"A callback function can be used to monitor convergence progress."
]
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"i = 0\n",
"def progress(x, y):\n",
" global i\n",
" plt.annotate(str(i), (x, y))\n",
" i += 1\n",
"sol = optimize.root(f, 1.0, method='broyden1', callback=progress)\n",
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"source": [
"#### Explicit Jacobian\n",
"Newton methods rely on the derivative. The default is to compute it using finite differences. Alternatively we can provide it using an explicit expression. This can be advantageous in terms of stability and efficiency, especially for functions that are expensive to evaluate."
]
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" fjac: array([[-1.]])\n",
" fun: array([ 1.11022302e-16])\n",
" message: 'The solution converged.'\n",
" nfev: 11\n",
" njev: 1\n",
" qtf: array([ 4.28423963e-12])\n",
" r: array([ 1.24648718])\n",
" status: 1\n",
" success: True\n",
" x: array([ 2.63127494])"
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"source": [
"### Multivariate functions\n",
"`f` and `x` can be vector valued. They must have the same dimension for root finding. Use minimize function under- and over-determined systems."
]
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"execution_count": 55,
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"f3 = lambda v: [np.cos(v[0] * v[1]), v[0]**2 - v[1]]"
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"text/plain": [
" fjac: array([[-0.50539205, 0.86288984],\n",
" [-0.86288984, -0.50539205]])\n",
" fun: array([ 2.91490756e-14, -1.79856130e-14])\n",
" message: 'The solution converged.'\n",
" nfev: 25\n",
" qtf: array([ -2.74510088e-09, -1.49387666e-09])\n",
" r: array([-2.70562263, -0.29402677, 1.49836439])\n",
" status: 1\n",
" success: True\n",
" x: array([-1.16244735, 1.35128385])"
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"source": [
"For systems with a very high number of dimensions, `method='krylov'` can be used to get an approximation for the inverse Jacobian. This should be much more efficient than a direct solution."
]
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"source": [
"## Fitting\n",
"### Linear fits\n",
"This not only applies to linear functions but anything that can be transformed into a function that linearly depends on the parameters."
]
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"source": [
"### Non-linear fits\n",
"A generic interface to non-linear least-squares fits is provided by `optimize.curve_fit`. It can also handle constraints on the parameters."
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" return a * np.exp(b * x)\n",
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"For the proper fit we need two exponentials.\n",
"\n",
"$$\\sinh x = \\frac{e^x - e^{-x}}{2}$$"
]
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{
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"[ 0.13407153 6.27733772 -0.08284892 -6.96303914]\n",
"[[ 2.02895750e-02 -1.59998745e-01 -4.53770902e-05 5.71764834e-04]\n",
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" return a * np.exp(b * x) + c * np.exp(d * x)\n",
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"print(cov)"
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"source": [
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"source": [
"## Integration\n",
"`scipy.integrate` contains routines for computing one or multidimensional integrals of functions or sampled data. It also provides an interface to many ODE integrators."
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"collapsed": true
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"outputs": [],
"source": [
"from scipy import integrate"
]
},
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"metadata": {
"slideshow": {
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}
},
"source": [
"### General-purpose function integrator\n",
"`quad` is an interface to the Fortran library QUADPACK. It is convenient when the integrand is available as a callable function."
]
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"execution_count": 65,
"metadata": {
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"source": [
"def f(x):\n",
" return np.sin(x) + np.sqrt(x)\n",
"def F(x): # analytic solution for verification\n",
" return - np.cos(x) + 2./3. * x**(1.5)"
]
},
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"execution_count": 66,
"metadata": {
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"outputs": [
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"data": {
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]
},
"execution_count": 66,
"metadata": {},
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}
],
"source": [
"integrate.quad(f,0.,2.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The return value is the result and an error estimate. You can set `full_output=True` to get more diagnostic output."
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
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"metadata": {},
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],
"source": [
"F(2) - F(0)"
]
},
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"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"One or both of the integration limits can be infinite.`"
]
},
{
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"execution_count": 68,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def g(x):\n",
" return np.exp(-x**2)"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
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"data": {
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"(1.7724538509055159, 1.4202636780944923e-08)"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"integrate.quad(g,-np.inf,np.inf)"
]
},
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"execution_count": 70,
"metadata": {},
"outputs": [
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"execution_count": 70,
"metadata": {},
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"source": [
"np.sqrt(np.pi)"
]
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"metadata": {
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"slide_type": "subslide"
}
},
"source": [
"### Sampled data\n",
"If just a fixed set of data samples is available, we can use `simps` to apply Simpson's rule to integrate the data."
]
},
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"execution_count": 71,
"metadata": {},
"outputs": [],
"source": [
"# create data sample\n",
"x = np.linspace(0.,2.,10)\n",
"y = f(x)"
]
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{
"cell_type": "code",
"execution_count": 72,
"metadata": {
"slideshow": {
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"outputs": [
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"data": {
"text/plain": [
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]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
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"source": [
"integrate.simps(y,x=x)"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
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"execution_count": 73,
"metadata": {},
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"source": [
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]
},
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"metadata": {
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"slide_type": "subslide"
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},
"source": [
"## Cumulative integration\n",
"Assume there is sampled data of a function $f(x)$ and you need to compute\n",
"$$\\int_0^x f(t)dt.$$\n",
"It is not efficient to compute the full integral for every point. Cumulative integration reuses the previous result."
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"y_int = integrate.cumtrapz(y, x=x, initial=0)\n",
"# cumulative trapezoidal rule on sampled data"
]
},
{
"cell_type": "code",
"execution_count": 75,
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"slide_type": "subslide"
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"### Ordinary differential equations\n",
"`integrate.ode` provides an interface to a variety of ODE integration methods. It allows us to set up the problem once and try many different integrators.\n",
"\n",
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"# sample the solution over an interval\n",
"x = np.linspace(0.01,4*np.pi)\n",
"y = [ode.integrate(t)[1] for t in x]\n",
"# Here we extract just the y component"
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"source": [
"plt.plot(x,y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## I/O\n",
"Many programming languages make use of special binary formats. `scipy.io` has methods for reading a few of these, for example Fortran unformatted output, Matlab `.mat` files, or IDL save files."
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from scipy import io"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Fortran unformatted output\n",
"Consider the following Fortran code\n",
"```fortran\n",
"program testout\n",
" implicit none\n",
" integer, parameter :: n=200\n",
" real :: a(n)\n",
" integer :: i\n",
" do i=1,n\n",
" a(i) = i\n",
" end do\n",
" open(42,file='fortranout',form='unformatted')\n",
" write(42) n\n",
" write(42) a\n",
" close(42)\n",
"end program\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([200], dtype=int32)"
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f=io.FortranFile('fortranout')\n",
"f.read_ints(dtype='i4')"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., 2., 3., 4., 5., 6., 7., 8., 9.,\n",
" 10., 11., 12., 13., 14., 15., 16., 17., 18.,\n",
" 19., 20., 21., 22., 23., 24., 25., 26., 27.,\n",
" 28., 29., 30., 31., 32., 33., 34., 35., 36.,\n",
" 37., 38., 39., 40., 41., 42., 43., 44., 45.,\n",
" 46., 47., 48., 49., 50., 51., 52., 53., 54.,\n",
" 55., 56., 57., 58., 59., 60., 61., 62., 63.,\n",
" 64., 65., 66., 67., 68., 69., 70., 71., 72.,\n",
" 73., 74., 75., 76., 77., 78., 79., 80., 81.,\n",
" 82., 83., 84., 85., 86., 87., 88., 89., 90.,\n",
" 91., 92., 93., 94., 95., 96., 97., 98., 99.,\n",
" 100., 101., 102., 103., 104., 105., 106., 107., 108.,\n",
" 109., 110., 111., 112., 113., 114., 115., 116., 117.,\n",
" 118., 119., 120., 121., 122., 123., 124., 125., 126.,\n",
" 127., 128., 129., 130., 131., 132., 133., 134., 135.,\n",
" 136., 137., 138., 139., 140., 141., 142., 143., 144.,\n",
" 145., 146., 147., 148., 149., 150., 151., 152., 153.,\n",
" 154., 155., 156., 157., 158., 159., 160., 161., 162.,\n",
" 163., 164., 165., 166., 167., 168., 169., 170., 171.,\n",
" 172., 173., 174., 175., 176., 177., 178., 179., 180.,\n",
" 181., 182., 183., 184., 185., 186., 187., 188., 189.,\n",
" 190., 191., 192., 193., 194., 195., 196., 197., 198.,\n",
" 199., 200.], dtype=float32)"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.read_reals(dtype='f4')"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Records involving more than one type can be read using the `read_record` method.\n",
"- You can specify endianness in the dtype like `'>f8'` (big endian double precision float) `'<i4'` (little endian 4-byte integer)\n",
"- Fortran arrays are column-major, numpy arrays are row-major by default. Loaded data is one-dimensional at first. Reshape it using Fortran ordering with\n",
" ```python\n",
" A = f.read_reals()\n",
" A.reshape((n,m),order='F')\n",
" ```\n",
"- FortranFile can also be used to write files by specifying `mode='w'` when opening."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Matlab\n",
"Matlab uses `.mat` files to save variables in binary form. Scipy has routines to load and save `.mat` files.\n",
"\n",
"Consider the following Matlab code:\n",
"```matlab\n",
"A=rand(300,300);\n",
"save test.mat A\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
"scrolled": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"{'A': array([[ 0.8937406 , 0.33096039, 0.27772644, ..., 0.43196036,\n",
" 0.11713621, 0.38315384],\n",
" [ 0.81624274, 0.85436018, 0.82194892, ..., 0.47457527,\n",
" 0.04818797, 0.54486213],\n",
" [ 0.94742332, 0.36182472, 0.62477771, ..., 0.76898039,\n",
" 0.07675434, 0.69719725],\n",
" ..., \n",
" [ 0.51246352, 0.82407083, 0.22070771, ..., 0.95076692,\n",
" 0.71071825, 0.01321776],\n",
" [ 0.579559 , 0.71837939, 0.71279853, ..., 0.2803856 ,\n",
" 0.56203512, 0.23141348],\n",
" [ 0.84749724, 0.37592622, 0.51177889, ..., 0.84780634,\n",
" 0.1806365 , 0.93363184]]),\n",
" '__globals__': [],\n",
" '__header__': 'MATLAB 5.0 MAT-file, Platform: GLNXA64, Created on: Thu Jun 15 11:10:58 2017',\n",
" '__version__': '1.0'}"
]
},
"execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"io.loadmat('test.mat')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Data are returned in a dictionary. `io.savemat` can be used to write Matlab data files."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### IDL\n",
"Consider a file saved with the follwing IDL code\n",
"```idl\n",
"A = [1,2,3]\n",
"save, /variables, 'A', filename='idltest.sav'\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'a': array([1, 2, 3], dtype=int16)}"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"io.readsav('idltest.sav')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Writing these files is not supported at the moment."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Sparse matrices\n",
"Scipy supports many operations on sparse matrices (matrices storing only the \"few\" non-zero entries). They occur in many problems, for example in discretisations of large systems of PDEs. It can convert between different sparse matrix formats and can make use of efficient algorithms on these matrices."
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from scipy import sparse"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [],
"source": [
"# example matrix\n",
"A = np.diag(np.arange(1,31, dtype='f8'))\n",
"A[12,15] = 47.\n",
"A[20,0] = -15."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Formats\n",
"Can be initialised using a normal (dense) matrix, a sparse matrix in another format, coordinate format `(data, (row_ind, col_ind))` or specifying the index vectors directly. The latter is really useful when reading sparse matrices from disk."
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"32"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Asp = sparse.csc_matrix(A) # Compressed Sparse Column format\n",
"Asp.nnz # number of non-zero elements"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 12,\n",
" 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], dtype=int32)"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Asp.indices"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18,\n",
" 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], dtype=int32)"
]
},
"execution_count": 94,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Asp.indptr"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., -15., 2., 3., 4., 5., 6., 7., 8., 9., 10.,\n",
" 11., 12., 13., 14., 15., 47., 16., 17., 18., 19., 20.,\n",
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]
},
"execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
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{
"cell_type": "code",
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"metadata": {
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}
},
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" [ -1.85676954e-17+0.j, 5.98793707e-17+0.j, 4.01901055e-17+0.j,\n",
" -1.63990400e-17+0.j, 5.98576255e-17+0.j, 8.36639811e-17+0.j],\n",
" [ 2.68024922e-17+0.j, 1.05802245e-16+0.j, -2.77803037e-17+0.j,\n",
" 1.98105969e-17+0.j, 1.72806807e-16+0.j, 5.88105598e-18+0.j],\n",
" [ 2.90901899e-16+0.j, 8.64351099e-16+0.j, -1.50171856e-16+0.j,\n",
" 5.58917143e-17+0.j, 6.10181858e-16+0.j, -6.65005692e-16+0.j],\n",
" [ -9.35908664e-17+0.j, -2.42766257e-16+0.j, 7.52241032e-17+0.j,\n",
" 1.26009114e-16+0.j, -2.51848038e-16+0.j, 2.47928680e-16+0.j],\n",
" [ -6.82740766e-17+0.j, -1.30783564e-16+0.j, 5.27331653e-17+0.j,\n",
" 3.15746447e-17+0.j, -1.20740691e-16+0.j, 8.60365491e-17+0.j],\n",
" [ 6.94819707e-17+0.j, 1.75468668e-16+0.j, -4.13958764e-17+0.j,\n",
" -3.17778213e-17+0.j, 1.50257629e-16+0.j, -1.08217172e-16+0.j],\n",
" [ -7.63078144e-17+0.j, -1.70079622e-16+0.j, -1.53565810e-17+0.j,\n",
" 1.86246723e-16+0.j, 2.31958061e-16+0.j, 1.04883764e-16+0.j],\n",
" [ -1.86531598e-18+0.j, -1.06411823e-17+0.j, -1.49134855e-17+0.j,\n",
" 8.69867943e-18+0.j, 1.36311501e-16+0.j, -3.44158371e-17+0.j],\n",
" [ -1.87518334e-19+0.j, -3.81551974e-18+0.j, 8.33630580e-19+0.j,\n",
" 1.89598277e-17+0.j, -1.39985307e-17+0.j, 2.43758385e-18+0.j],\n",
" [ -1.09869707e-16+0.j, -3.40226845e-16+0.j, -8.76048472e-17+0.j,\n",
" 7.70633343e-16+0.j, 3.67368526e-16+0.j, -3.32920613e-16+0.j],\n",
" [ 3.21389630e-16+0.j, 6.93240515e-16+0.j, -2.42840736e-16+0.j,\n",
" -2.43570461e-16+0.j, -9.62349526e-16+0.j, 5.42981617e-16+0.j],\n",
" [ -5.75684000e-17+0.j, -1.51282308e-16+0.j, 1.56575581e-16+0.j,\n",
" -4.20720366e-16+0.j, 1.47547477e-16+0.j, 2.13585838e-17+0.j],\n",
" [ 4.12884981e-18+0.j, -1.46797179e-16+0.j, -1.88529807e-16+0.j,\n",
" -2.78963644e-18+0.j, 3.01402225e-17+0.j, 2.75500338e-16+0.j],\n",
" [ 1.44740853e-17+0.j, 7.00073290e-16+0.j, 1.39045573e-15+0.j,\n",
" -9.46746472e-16+0.j, -7.78105975e-16+0.j, -1.31946518e-14+0.j],\n",
" [ -1.16833440e-16+0.j, -6.79926317e-16+0.j, -1.07740050e-16+0.j,\n",
" 4.91186952e-15+0.j, -4.03982403e-14+0.j, 1.00000000e+00+0.j],\n",
" [ 1.59299700e-16+0.j, 6.53208092e-16+0.j, 6.49545522e-16+0.j,\n",
" -3.84414722e-15+0.j, 1.00000000e+00+0.j, 3.99961749e-14+0.j],\n",
" [ 6.69969180e-17+0.j, 8.69503037e-16+0.j, 3.64118458e-15+0.j,\n",
" -1.00000000e+00+0.j, -2.58257348e-15+0.j, 5.61778186e-15+0.j],\n",
" [ -6.46835016e-16+0.j, -1.47243329e-14+0.j, -1.00000000e+00+0.j,\n",
" -2.86491005e-15+0.j, -2.06938457e-16+0.j, 3.20928928e-16+0.j],\n",
" [ 1.38083989e-15+0.j, 1.00000000e+00+0.j, -1.59070177e-14+0.j,\n",
" 1.57083199e-15+0.j, -5.07605097e-16+0.j, 2.03893731e-16+0.j],\n",
" [ 1.00000000e+00+0.j, -8.21887901e-16+0.j, -1.28546314e-15+0.j,\n",
" 4.05718446e-16+0.j, -2.27962457e-16+0.j, 7.38738606e-16+0.j]]))"
]
},
"execution_count": 101,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sparse.linalg.eigs(Asp)\n",
"# compute a few eigenvalues and vectors"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Iterative solvers\n",
"For very large sparse matrices it is not efficient and sometimes not even possible (due to memory) to compute the inverse of a matrix. There are very efficient iterative solution methods to linear systems. Some work on general matrices (GMRES, BiCGSTAB, ...), some only on symmetric matrices (CG, ...). Scipy implements some of these methods. "
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"b = np.ones(Asp.shape[0]) # RHS"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"(array([ 0.99999368, 0.50000344, 0.33332747, 0.25000626, 0.19999668,\n",
" 0.16666672, 0.14285699, 0.12500163, 0.11111062, 0.09999859,\n",
" 0.0909096 , 0.08333444, -0.14904021, 0.07142836, 0.06666668,\n",
" 0.06250001, 0.0588236 , 0.05555573, 0.05263146, 0.04999979,\n",
" 0.76190009, 0.04545487, 0.04347809, 0.04166662, 0.0399999 ,\n",
" 0.03846183, 0.03703678, 0.03571439, 0.03448273, 0.03333334]), 0)"
]
},
"execution_count": 103,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# GMRES method for non-symmetric matrices\n",
"sparse.linalg.gmres(Asp, b)"
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(array([ 0.99997922, 0.50001088, 0.33333347, 0.25000016, 0.20000001,\n",
" 0.16666667, 0.14285715, 0.12499997, 0.11111107, 0.1 ,\n",
" 0.09090909, 0.08333334, -0.14903857, 0.07142858, 0.0666667 ,\n",
" 0.06249999, 0.0588235 , 0.05555555, 0.05263157, 0.04999999,\n",
" 0.76188918, 0.04545455, 0.04347822, 0.04166679, 0.04000058,\n",
" 0.03846133, 0.03703679, 0.03571399, 0.03448278, 0.03333268]), 0)"
]
},
"execution_count": 104,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sparse.linalg.bicgstab(Asp, b)"
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1. , 0.5 , 0.33333333, 0.25 , 0.2 ,\n",
" 0.16666667, 0.14285714, 0.125 , 0.11111111, 0.1 ,\n",
" 0.09090909, 0.08333333, -0.14903846, 0.07142857, 0.06666667,\n",
" 0.0625 , 0.05882353, 0.05555556, 0.05263158, 0.05 ,\n",
" 0.76190476, 0.04545455, 0.04347826, 0.04166667, 0.04 ,\n",
" 0.03846154, 0.03703704, 0.03571429, 0.03448276, 0.03333333])"
]
},
"execution_count": 105,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.linalg.solve(A, b)"
]
}
],
"metadata": {
"celltoolbar": "Slideshow",
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"display_name": "Python 3",
"language": "python",
"name": "python3"
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"language_info": {
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"name": "ipython",
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