numpy-100/100 Numpy exercises.rst

100 numpy exercises
===================

This is a collection of exercises that have been collected in the numpy mailing
list, on stack overflow and in the numpy documentation. I've also created some
to reach the 100 limit. The goal of this collection is to offer a quick
reference for both old and new users but also to provide a set of exercices for
those who teach.

If you find an error or think you've a better way to solve some of them, feel
free to open an issue at https://github.com/rougier/numpy-100


#. Import the numpy package under the name ``np`` (★☆☆) 

   .. code-block:: python

      import numpy as np


#. Print the numpy version and the configuration (★☆☆) 

   .. code-block:: python

      print(np.__version__)
      np.show_config()


#. Create a null vector of size 10 (★☆☆) 

   .. code-block:: python

      Z = np.zeros(10)
      print(Z)

#. How to find the memory size of any array (★☆☆) 

   .. code-block:: python

      Z = np.zeros((10,10))
      print("%d bytes" % (Z.size * Z.itesize))

                   
#. How to get the documentation of the numpy add function from the command line? (★☆☆) 

   .. code-block:: bash

      python -c "import numpy; numpy.info(numpy.add)"


#. Create a null vector of size 10 but the fifth value which is 1 (★☆☆) 

   .. code-block:: python

      Z = np.zeros(10)
      Z[4] = 1
      print(Z)


#. Create a vector with values ranging from 10 to 49 (★☆☆) 

   .. code-block:: python

      Z = np.arange(10,50)
      print(Z)


#. Reverse a vector (first element becomes last) (★☆☆) 

   .. code-block:: python

      Z = np.arange(50)
      Z = Z[::-1]


#. Create a 3x3 matrix with values ranging from 0 to 8 (★☆☆) 

   .. code-block:: python

      Z = np.arange(9).reshape(3,3)
      print(Z)


#. Find indices of non-zero elements from [1,2,0,0,4,0] (★☆☆) 

   .. code-block:: python

      nz = np.nonzero([1,2,0,0,4,0])
      print(nz)


#. Create a 3x3 identity matrix (★☆☆) 

   .. code-block:: python

      Z = np.eye(3)
      print(Z)


#. Create a 3x3x3 array with random values (★☆☆) 

   .. code-block:: python

      Z = np.random.random((3,3,3))
      print(Z)


#. Create a 10x10 array with random values and find the minimum and maximum values (★☆☆) 

   .. code-block:: python

      Z = np.random.random((10,10))
      Zmin, Zmax = Z.min(), Z.max()
      print(Zmin, Zmax)

      
#. Create a random vector of size 30 and find the mean value  (★☆☆) 

   .. code-block:: python

      Z = np.random.random(30)
      m = Z.mean()
      print(m)

      
#. Create a 2d array with 1 on the border and 0 inside (★☆☆) 

   .. code-block:: python

      Z = np.ones((10,10))
      Z[1:-1,1:-1] = 0

#. How to add a border (filled with 0's) around an existing array ? (★☆☆)
   
   .. code-block:: python

      Z = np.ones((5,5))
      Z = np.pad(Z, pad_width=1, mode='constant', constant_values=0)

#. What is the result of the following expression? (★☆☆)

   .. code-block:: python

      0 * np.nan
      np.nan == np.nan
      np.inf > np.nan
      np.nan - np.nan
      0.3 == 3 * 0.1

      
#. Create a 5x5 matrix with values 1,2,3,4 just below the diagonal (★☆☆) 

   .. code-block:: python

      Z = np.diag(1+np.arange(4),k=-1)
      print(Z)
      

#. Create a 8x8 matrix and fill it with a checkerboard pattern (★☆☆) 

   .. code-block:: python

      Z = np.zeros((8,8),dtype=int)
      Z[1::2,::2] = 1
      Z[::2,1::2] = 1
      print(Z)

#. Consider a (6,7,8) shape array, what is the index (x,y,z) of the 100th element?

   .. code-block:: python

      print(np.unravel_index(100,(6,7,8)))

#. Create a checkerboard 8x8 matrix using the tile function (★☆☆) 

   .. code-block:: python

      Z = np.tile( np.array([[0,1],[1,0]]), (4,4))
      print(Z)


#. Normalize a 5x5 random matrix (★☆☆) 

   .. code-block:: python

      Z = np.random.random((5,5))
      Zmax, Zmin = Z.max(), Z.min()
      Z = (Z - Zmin)/(Zmax - Zmin)
      print(Z)

#. Create a custom dtype that describes a color as four unisgned bytes (RGBA) (★☆☆) 

   .. code-block:: python

      color = np.dtype([("r", np.ubyte, 1),
                        ("g", np.ubyte, 1),
                        ("b", np.ubyte, 1),
                        ("a", np.ubyte, 1)])

#. Multiply a 5x3 matrix by a 3x2 matrix (real matrix product) (★☆☆) 

   .. code-block:: python

      Z = np.dot(np.ones((5,3)), np.ones((3,2)))
      print(Z)

#. Given a 1D array, negate all elements which are between 3 and 8, in place. (★☆☆)
   
   .. code-block:: python

      # Author: Evgeni Burovski

      Z = np.arange(11)
      Z[(3 < Z) & (Z <= 8)] *= -1

#. What is the output of the following script? (★☆☆)
   
   .. code-block:: python

      # Author: Jake VanderPlas

      print(sum(range(5),-1))
      from numpy import *
      print(sum(range(5),-1))

#. Consider an integer vector Z, which of these expressions are legal? (★☆☆)
   
   .. code-block:: python

      Z**Z
      2 << Z >> 2
      Z <- Z
      1j*Z
      Z/1/1
      Z<Z>Z

#. What are the result of the following expressions?

   .. code-block:: python

      np.array(0) / np.array(0)
      np.array(0) // np.array(0)
      np.array([np.nan]).astype(int).astype(float)
      
#. How to round away from zero a float array ? (★☆☆)
   
   .. code-block:: python

      # Author: Charles R Harris

      Z = np.random.uniform(-10,+10,10)
      print (np.trunc(Z + np.copysign(0.5, Z)))


#. How to find common values between two arrays? (★☆☆)

   .. code-block:: python

      Z1 = np.random.randint(0,10,10)
      Z2 = np.random.randint(0,10,10)
      print(np.intersect1d(Z1,Z2))

#. How to ignore all numpy warnings (not recommended)? (★☆☆)
   
   .. code-block:: python

      # Suicide mode on
      defaults = np.seterr(all="ignore")
      Z = np.ones(1)/0

      # Back to sanity
      np.seterr(**defaults)
      
   
#. Is the following expressions true? (★☆☆)

   .. code-block:: python

      np.sqrt(-1) == np.emath.sqrt(-1)

#. How to get the dates of yesterday, today and tomorrow? (★☆☆) 

   .. code-block:: python

      yesterday = np.datetime64('today', 'D') - np.timedelta64(1, 'D')
      today     = np.datetime64('today', 'D')
      tomorrow  = np.datetime64('today', 'D') + np.timedelta64(1, 'D')

      
#. How to get all the dates corresponding to the month of July 2016? (★★☆) 

   .. code-block:: python

      Z = np.arange('2016-07', '2016-08', dtype='datetime64[D]')
      print(Z)
      

#. How to compute ((A+B)*(-A/2)) in place (without copy)? (★★☆)

   .. code-block:: python

      A = np.ones(3)*1
      B = np.ones(3)*2
      C = np.ones(3)*3
      np.add(A,B,out=B)
      np.divide(A,2,out=A)
      np.negative(A,out=A)
      np.multiply(A,B,out=A)
      
#. Extract the integer part of a random array using 5 different methods (★★☆)

   .. code-block:: python

      Z = np.random.uniform(0,10,10)

      print (Z - Z%1)
      print (np.floor(Z))
      print (np.ceil(Z)-1)
      print (Z.astype(int))
      print (np.trunc(Z))


#. Create a 5x5 matrix with row values ranging from 0 to 4 (★★☆) 

   .. code-block:: python

    Z = np.zeros((5,5))
    Z += np.arange(5)
    print(Z)

    
#. Consider a generator function that generates 10 integers and use it to build an
   array (★☆☆) 

   .. code-block:: python

      def generate():
          for x in xrange(10):
              yield x
      Z = np.fromiter(generate(),dtype=float,count=-1)
      print(Z)


#. Create a vector of size 10 with values ranging from 0 to 1, both excluded (★★☆) 

   .. code-block:: python

    Z = np.linspace(0,1,12,endpoint=True)[1:-1]
    print(Z)


#. Create a random vector of size 10 and sort it (★★☆) 

   .. code-block:: python

    Z = np.random.random(10)
    Z.sort()
    print(Z)


#. How to sum a small array faster than np.sum? (★★☆) 

   .. code-block:: python

      # Author: Evgeni Burovski
   
      Z = np.arange(10)
      np.add.reduce(Z)

      
#. Consider two random array A anb B, check if they are equal  (★★☆) 

   .. code-block:: python

      A = np.random.randint(0,2,5)
      B = np.random.randint(0,2,5)
      equal = np.allclose(A,B)
      print(equal)



#. Make an array immutable (read-only) (★★☆) 

   .. code-block:: python

      Z = np.zeros(10)
      Z.flags.writeable = False
      Z[0] = 1


#. Consider a random 10x2 matrix representing cartesian coordinates, convert
   them to polar coordinates (★★☆) 

   .. code-block:: python

      Z = np.random.random((10,2))
      X,Y = Z[:,0], Z[:,1]
      R = np.sqrt(X**2+Y**2)
      T = np.arctan2(Y,X)
      print(R)
      print(T)


#. Create random vector of size 10 and replace the maximum value by 0 (★★☆) 

   .. code-block:: python

    Z = np.random.random(10)
    Z[Z.argmax()] = 0
    print(Z)


#. Create a structured array with ``x`` and ``y`` coordinates covering the
   [0,1]x[0,1] area (★★☆) 

   .. code-block:: python

      Z = np.zeros((10,10), [('x',float),('y',float)])
      Z['x'], Z['y'] = np.meshgrid(np.linspace(0,1,10),
                                   np.linspace(0,1,10))
      print(Z)


#. Given two arrays, X and Y, construct the Cauchy matrix C (Cij = 1/(xi - yj))

   .. code-block:: python

      # Author: Evgeni Burovski
                   
      X = np.arange(8)
      Y = X + 0.5
      C = 1.0 / np.subtract.outer(X, Y)
      print(np.linalg.det(C))


#. Print the minimum and maximum representable value for each numpy scalar type (★★☆) 

   .. code-block:: python

      for dtype in [np.int8, np.int32, np.int64]:
         print(np.iinfo(dtype).min)
         print(np.iinfo(dtype).max)
      for dtype in [np.float32, np.float64]:
         print(np.finfo(dtype).min)
         print(np.finfo(dtype).max)
         print(np.finfo(dtype).eps)

#. How to print all the values of an array?  (★★☆)

   .. code-block:: python

      np.set_printoptions(threshold=np.nan)
      Z = np.zeros((25,25))
      print(Z)


#. How to find the closest value (to a given scalar) in an array?  (★★☆)
   
   .. code-block:: python

      Z = np.arange(100)
      v = np.random.uniform(0,100)
      index = (np.abs(Z-v)).argmin()
      print(Z[index])

#. Create a structured array representing a position (x,y) and a color (r,g,b) (★★☆)

   .. code-block:: python

      Z = np.zeros(10, [ ('position', [ ('x', float, 1),
                                        ('y', float, 1)]),
                         ('color',    [ ('r', float, 1),
                                        ('g', float, 1),
                                        ('b', float, 1)])])
     print(Z)


#. Consider a random vector with shape (100,2) representing coordinates, find
   point by point distances (★★☆) 

   .. code-block:: python

      Z = np.random.random((10,2))
      X,Y = np.atleast_2d(Z[:,0]), np.atleast_2d(Z[:,1])
      D = np.sqrt( (X-X.T)**2 + (Y-Y.T)**2)
      print(D)

      # Much faster with scipy
      import scipy
      # Thanks Gavin Heverly-Coulson (#issue 1)
      import scipy.spatial
      
      Z = np.random.random((10,2))
      D = scipy.spatial.distance.cdist(Z,Z)
      print(D)


#. How to convert a float (32 bits) array into an integer (32 bits) in place?

   .. code-block:: python

      Z = np.arange(10, dtype=np.int32)
      Z = Z.astype(np.float32, copy=False)
                   
   
#. How to read the following file? (★★☆)

   .. code-block:: python

      # File content:
      # -------------
      1,2,3,4,5
      6,,,7,8
      ,,9,10,11
      # -------------

      Z = np.genfromtxt("missing.dat", delimiter=",")


#. What is the equivalent of `enumerate` for numpy arrays? (★★☆)

   .. code-block:: python

      Z = np.arange(9).reshape(3,3)
      for index, value in np.ndenumerate(Z):
          print(index, value)
      for index in np.ndindex(Z.shape):
          print(index, Z[index])
                   
         
#. Generate a generic 2D Gaussian-like array (★★☆) 

   .. code-block:: python

      X, Y = np.meshgrid(np.linspace(-1,1,10), np.linspace(-1,1,10))
      D = np.sqrt(X*X+Y*Y)
      sigma, mu = 1.0, 0.0
      G = np.exp(-( (D-mu)**2 / ( 2.0 * sigma**2 ) ) )
      print(G)

#. How to randomly place p elements in a 2D array? (★★☆)

   .. code-block:: python

      # Author: Divakar
       
      n = 10
      p = 3
      Z = np.zeros((n,n))
      np.put(Z, np.random.choice(range(n*n), p, replace=False),1)

#. Subtract the mean of each row of a matrix (★★☆) 

   .. code-block:: python

      # Author: Warren Weckesser

      X = np.random.rand(5, 10)

      # Recent versions of numpy
      Y = X - X.mean(axis=1, keepdims=True)

      # Older versions of numpy
      Y = X - X.mean(axis=1).reshape(-1, 1)

#. How to I sort an array by the nth column? (★★☆) 

   .. code-block:: python

      # Author: Steve Tjoa
      
      Z = np.random.randint(0,10,(3,3))
      print(Z)
      print(Z[Z[:,1].argsort()])
   

#. How to tell if a given 2D array has null columns? (★★☆) 

   .. code-block:: python

      # Author: Warren Weckesser

      Z = np.random.randint(0,3,(3,10))
      print((~Z.any(axis=0)).any())

      
#. Find the nearest value from a given value in an array (★★☆) 

   .. code-block:: python

      Z = np.random.uniform(0,1,10)
      z = 0.5
      m = Z.flat[np.abs(Z - z).argmin()]
      print(m)

#. Considering two arrays with shape (1,3) and (3,1), how to compute their sum
   using an iterator? (★★☆)

   .. code-block:: python

      A = np.arange(3).reshape(3,1)
      B = np.arange(3).reshape(1,3)
      it = np.nditer([A,B,None]):
      for x,y,z in it: z[...] = x + y
      C = it.operands[2]


#. Create an array class that has a `name` attribute (★★☆)

   .. code-block:: python

      class NamedArray(np.ndarray):
          def __new__(cls, array, name="no name"):
              obj = np.asarray(array).view(cls)
              obj.name = name
              return obj
          def __array_finalize__(self, obj):
              if obj is None: return
              self.info = getattr(obj, 'name', "no name")

      Z = NamedArray(np.arange(10), "range_10")
      print (Z.name)



#. Consider a given vector, how to add 1 to each element indexed by a second
   vector (be careful with repeated indices)? (★★★) 

   .. code-block:: python

      # Author: Brett Olsen

      Z = np.ones(10)
      I = np.random.randint(0,len(Z),20)
      Z += np.bincount(I, minlength=len(Z))
      print(Z)


#. How to accumulate elements of a vector (X) to an array (F) based on an index
   list (I)? (★★★) 

   .. code-block:: python

      # Author: Alan G Isaac

      X = [1,2,3,4,5,6]
      I = [1,3,9,3,4,1]
      F = np.bincount(I,X)
      print(F)


#. Considering a (w,h,3) image of (dtype=ubyte), compute the number of unique
   colors (★★★) 

   .. code-block:: python

      # Author: Nadav Horesh

      w,h = 16,16
      I = np.random.randint(0,2,(h,w,3)).astype(np.ubyte)
      F = I[...,0]*256*256 + I[...,1]*256 +I[...,2]
      n = len(np.unique(F))
      print(np.unique(I))


#. Considering a four dimensions array, how to get sum over the last two axis
   at once? (★★★)

   .. code-block:: python

      A = np.random.randint(0,10,(3,4,3,4))
      sum = A.reshape(A.shape[:-2] + (-1,)).sum(axis=-1)
      print(sum)


#. Considering a one-dimensional vector D, how to compute means of subsets of D
   using a vector S of same size describing subset indices? (★★★) 

   .. code-block:: python

      # Author: Jaime Fernández del Río

      D = np.random.uniform(0,1,100)
      S = np.random.randint(0,10,100)
      D_sums = np.bincount(S, weights=D)
      D_counts = np.bincount(S)
      D_means = D_sums / D_counts
      print(D_means)


#. How to get the diagonal of a dot product? (★★★) 

   .. code-block:: python

      # Author: Mathieu Blondel
                   
      # Slow version  
      np.diag(np.dot(A, B))

      # Fast version
      np.sum(A * B.T, axis=1)

      # Faster version
      np.einsum("ij,ji->i", A, B).


#. Consider the vector [1, 2, 3, 4, 5], how to build a new vector with 3
   consecutive zeros interleaved between each value?  (★★★) 

   .. code-block:: python

      # Author: Warren Weckesser

      Z = np.array([1,2,3,4,5])
      nz = 3
      Z0 = np.zeros(len(Z) + (len(Z)-1)*(nz))
      Z0[::nz+1] = Z
      print(Z0)


#. Consider an array of dimension (5,5,3), how to mulitply it by an array with
   dimensions (5,5)?  (★★★) 

   .. code-block:: python

      A = np.ones((5,5,3))
      B = 2*np.ones((5,5))
      print(A * B[:,:,None])


#. How to swap two rows of an array? (★★★) 

   .. code-block:: python

      # Author: Eelco Hoogendoorn

      A = np.arange(25).reshape(5,5)
      A[[0,1]] = A[[1,0]]
      print(A)


#. Consider a set of 10 triplets describing 10 triangles (with shared
   vertices), find the set of unique line segments composing all the triangles (★★★) 

   .. code-block:: python

      # Author: Nicolas P. Rougier

      faces = np.random.randint(0,100,(10,3))
      F = np.roll(faces.repeat(2,axis=1),-1,axis=1)
      F = F.reshape(len(F)*3,2)
      F = np.sort(F,axis=1)
      G = F.view( dtype=[('p0',F.dtype),('p1',F.dtype)] )
      G = np.unique(G)
      print(G)


#. Given an array C that is a bincount, how to produce an array A such that
   np.bincount(A) == C?  (★★★) 

   .. code-block:: python

     # Author: Jaime Fernández del Río

     C = np.bincount([1,1,2,3,4,4,6])
     A = np.repeat(np.arange(len(C)), C)
     print(A)

     
#. How to compute averages using a sliding window over an array? (★★★) 

   .. code-block:: python

      # Author: Jaime Fernández del Río

      def moving_average(a, n=3) :
          ret = np.cumsum(a, dtype=float)
          ret[n:] = ret[n:] - ret[:-n]
          return ret[n - 1:] / n
      Z = np.arange(20)
      print(moving_average(Z, n=3))

#. Consider a one-dimensional array Z, build a two-dimensional array whose
   first row is (Z[0],Z[1],Z[2]) and each subsequent row is shifted by 1 (last
   row should be (Z[-3],Z[-2],Z[-1]) (★★★) 

   .. code-block:: python

      # Author: Joe Kington / Erik Rigtorp
      from numpy.lib import stride_tricks

      def rolling(a, window):
          shape = (a.size - window + 1, window)
          strides = (a.itemsize, a.itemsize)
          return stride_tricks.as_strided(a, shape=shape, strides=strides)
      Z = rolling(np.arange(10), 3)
      print(Z)


#. How to negate a boolean, or to change the sign of a float inplace? (★★★) 

   .. code-block:: python

      # Author: Nathaniel J. Smith

      Z = np.random.randint(0,2,100)
      np.logical_not(arr, out=arr)

      Z = np.random.uniform(-1.0,1.0,100)
      np.negative(arr, out=arr)



#. Consider 2 sets of points P0,P1 describing lines (2d) and a point p, how to
   compute distance from p to each line i (P0[i],P1[i])? (★★★) 

   .. code-block:: python

      def distance(P0, P1, p):
          T = P1 - P0
          L = (T**2).sum(axis=1)
          U = -((P0[:,0]-p[...,0])*T[:,0] + (P0[:,1]-p[...,1])*T[:,1]) / L
          U = U.reshape(len(U),1)
          D = P0 + U*T - p
          return np.sqrt((D**2).sum(axis=1))

      P0 = np.random.uniform(-10,10,(10,2))
      P1 = np.random.uniform(-10,10,(10,2))
      p  = np.random.uniform(-10,10,( 1,2))
      print(distance(P0, P1, p))


#. Consider 2 sets of points P0,P1 describing lines (2d) and a set of points P,
   how to compute distance from each point j (P[j]) to each line i (P0[i],P1[i])? (★★★) 

   .. code-block:: python

      # Author: Italmassov Kuanysh
      # based on distance function from previous question
      P0 = np.random.uniform(-10, 10, (10,2))
      P1 = np.random.uniform(-10,10,(10,2))
      p = np.random.uniform(-10, 10, (10,2))
      print np.array([distance(P0,P1,p_i) for p_i in p])

#. Consider an arbitrary array, write a function that extract a subpart with a
   fixed shape and centered on a given element (pad with a ``fill`` value when
   necessary)  (★★★) 

   .. code:: python

      # Author: Nicolas Rougier

      Z = np.random.randint(0,10,(10,10))
      shape = (5,5)
      fill  = 0
      position = (1,1)

      R = np.ones(shape, dtype=Z.dtype)*fill
      P  = np.array(list(position)).astype(int)
      Rs = np.array(list(R.shape)).astype(int)
      Zs = np.array(list(Z.shape)).astype(int)

      R_start = np.zeros((len(shape),)).astype(int)
      R_stop  = np.array(list(shape)).astype(int)
      Z_start = (P-Rs//2)
      Z_stop  = (P+Rs//2)+Rs%2

      R_start = (R_start - np.minimum(Z_start,0)).tolist()
      Z_start = (np.maximum(Z_start,0)).tolist()
      R_stop = np.maximum(R_start, (R_stop - np.maximum(Z_stop-Zs,0))).tolist()
      Z_stop = (np.minimum(Z_stop,Zs)).tolist()

      r = [slice(start,stop) for start,stop in zip(R_start,R_stop)]
      z = [slice(start,stop) for start,stop in zip(Z_start,Z_stop)]
      R[r] = Z[z]
      print(Z)
      print(R)


#. Consider an array Z = [1,2,3,4,5,6,7,8,9,10,11,12,13,14], how to generate an
   array R = [[1,2,3,4], [2,3,4,5], [3,4,5,6], ..., [11,12,13,14]]? (★★★) 

   .. code-block:: python

      # Author: Stefan van der Walt

      Z = np.arange(1,15,dtype=uint32)
      R = stride_tricks.as_strided(Z,(11,4),(4,4))
      print(R)

#. Compute a matrix rank (★★★)

   .. code-block:: python

      # Author: Stefan van der Walt
    
      Z = np.random.uniform(0,1,(10,10))
      U, S, V = np.linalg.svd(Z) # Singular Value Decomposition
      rank = np.sum(S > 1e-10)

#. How to find the most frequent value in an array?

   .. code-block:: python

      Z = np.random.randint(0,10,50)
      print(np.bincount(Z).argmax())
      
#. Extract all the contiguous 3x3 blocks from a random 10x10 matrix (★★★) 

   .. code-block:: python

      # Author: Chris Barker

      Z = np.random.randint(0,5,(10,10))
      n = 3
      i = 1 + (Z.shape[0]-3)
      j = 1 + (Z.shape[1]-3)
      C = stride_tricks.as_strided(Z, shape=(i, j, n, n), strides=Z.strides + Z.strides)
      print(C)


#. Create a 2D array subclass such that Z[i,j] == Z[j,i] (★★★) 

   .. code-block:: python

      # Author: Eric O. Lebigot
      # Note: only works for 2d array and value setting using indices

      class Symetric(np.ndarray):
          def __setitem__(self, (i,j), value):
              super(Symetric, self).__setitem__((i,j), value)
              super(Symetric, self).__setitem__((j,i), value)

      def symetric(Z):
          return np.asarray(Z + Z.T - np.diag(Z.diagonal())).view(Symetric)

      S = symetric(np.random.randint(0,10,(5,5)))
      S[2,3] = 42
      print(S)

#. Consider a set of p matrices wich shape (n,n) and a set of p vectors with shape (n,1).
   How to compute the sum of of the p matrix products at once? (result has shape (n,1)) (★★★) 

   .. code-block:: python

      # Author: Stefan van der Walt

      p, n = 10, 20
      M = np.ones((p,n,n))
      V = np.ones((p,n,1))
      S = np.tensordot(M, V, axes=[[0, 2], [0, 1]])
      print(S)

      # It works, because:
      # M is (p,n,n)
      # V is (p,n,1)
      # Thus, summing over the paired axes 0 and 0 (of M and V independently),
      # and 2 and 1, to remain with a (n,1) vector.


#. Consider a 16x16 array, how to get the block-sum (block size is 4x4)? (★★★) 

   .. code-block:: python

      # Author: Robert Kern
                         
      Z = np.ones(16,16)
      k = 4
      S = np.add.reduceat(np.add.reduceat(Z, np.arange(0, Z.shape[0], k), axis=0),
                                             np.arange(0, Z.shape[1], k), axis=1)
                   

#. How to implement the Game of Life using numpy arrays? (★★★) 

   .. code-block:: python

      # Author: Nicolas Rougier

      def iterate(Z):
          # Count neighbours
          N = (Z[0:-2,0:-2] + Z[0:-2,1:-1] + Z[0:-2,2:] +
               Z[1:-1,0:-2]                + Z[1:-1,2:] +
               Z[2:  ,0:-2] + Z[2:  ,1:-1] + Z[2:  ,2:])

          # Apply rules
          birth = (N==3) & (Z[1:-1,1:-1]==0)
          survive = ((N==2) | (N==3)) & (Z[1:-1,1:-1]==1)
          Z[...] = 0
          Z[1:-1,1:-1][birth | survive] = 1
          return Z

      Z = np.random.randint(0,2,(50,50))
      for i in range(100): Z = iterate(Z)

#. How to get the n largest values of an array (★★★)

   .. code-block:: python

      Z = np.arange(10000)
      np.random.shuffle(Z)
      n = 5

      # Slow
      print (Z[np.argsort(Z)[-n:]])
      
      # Fast
      print (Z[np.argpartition(-Z,n)[:n]])
      

#. Given an arbitrary number of vectors, build the cartesian product (every
   combinations of every item) (★★★)
   
   .. code-block:: python

      # Author: Stefan Van der Walt

      def cartesian(arrays):
          arrays = [np.asarray(a) for a in arrays]
          shape = (len(x) for x in arrays)

          ix = np.indices(shape, dtype=int)
          ix = ix.reshape(len(arrays), -1).T

          for n, arr in enumerate(arrays):
              ix[:, n] = arrays[n][ix[:, n]]

          return ix

      print (cartesian(([1, 2, 3], [4, 5], [6, 7])))


#. How to create a record array from a regular array? (★★★) 
   
   .. code-block:: python

      Z = np.array([("Hello", 2.5, 3),
                    ("World", 3.6, 2)])
      R = np.core.records.fromarrays(Z.T, 
                                     names='col1, col2, col3',
                                     formats = 'S8, f8, i8')

#. Consider a large vector Z, compute Z to the power of 3 using 3 different
   methods (★★★)
   
   .. code-block:: python

      Author: Ryan G.
                   
      x = np.random.rand(5e7)

      %timeit np.power(x,3)
      1 loops, best of 3: 574 ms per loop

      %timeit x*x*x
      1 loops, best of 3: 429 ms per loop

      %timeit np.einsum('i,i,i->i',x,x,x)
      1 loops, best of 3: 244 ms per loop
 
                   
#. Consider two arrays A and B of shape (8,3) and (2,2). How to find rows of A
   that contain elements of each row of B regardless of the order of the elements
   in B? (★★★) 

   .. code-block:: python

      # Author: Gabe Schwartz

      A = np.random.randint(0,5,(8,3))
      B = np.random.randint(0,5,(2,2))

      C = (A[..., np.newaxis, np.newaxis] == B)
      rows = (C.sum(axis=(1,2,3)) >= B.shape[1]).nonzero()[0]
      print(rows)


#. Considering a 10x3 matrix, extract rows with unequal values (e.g. [2,2,3]) (★★★) 

   .. code-block:: python

      # Author: Robert Kern

      Z = np.random.randint(0,5,(10,3))
      E = np.logical_and.reduce(Z[:,1:] == Z[:,:-1], axis=1)
      U = Z[~E]
      print(Z)
      print(U)


#. Convert a vector of ints into a matrix binary representation (★★★) 

   .. code-block:: python

      # Author: Warren Weckesser

      I = np.array([0, 1, 2, 3, 15, 16, 32, 64, 128])
      B = ((I.reshape(-1,1) & (2**np.arange(8))) != 0).astype(int)
      print(B[:,::-1])

      # Author: Daniel T. McDonald

      I = np.array([0, 1, 2, 3, 15, 16, 32, 64, 128], dtype=np.uint8)
      print(np.unpackbits(I[:, np.newaxis], axis=1))


#. Given a two dimensional array, how to extract unique rows? (★★★) 

   .. code-block:: python

      # Author: Jaime Fernández del Río

      Z = np.random.randint(0,2,(6,3))
      T = np.ascontiguousarray(Z).view(np.dtype((np.void, Z.dtype.itemsize * Z.shape[1])))
      _, idx = np.unique(T, return_index=True)
      uZ = Z[idx]
      print(uZ)

#. Considering 2 vectors A & B, write the einsum equivalent of inner, outer,
   sum, and mul function (★★★) 


   .. code-block:: python

      # Author: Alex Riley
      # Make sure to read: http://ajcr.net/Basic-guide-to-einsum/

      np.einsum('i->', A)       # np.sum(A)
      np.einsum('i,i->i', A, B) # A * B
      np.einsum('i,i', A, B)	# np.inner(A, B)
      np.einsum('i,j', A, B)    # np.outer(A, B)


#. Considering a path described by two vectors (X,Y), how to sample it using
   equidistant samples (★★★)?

   .. code-block:: python

      # Author: Bas Swinckels

      phi = np.arange(0, 10*np.pi, 0.1)
      a = 1
      x = a*phi*np.cos(phi)
      y = a*phi*np.sin(phi)

      dr = (np.diff(x)**2 + np.diff(y)**2)**.5 # segment lengths
      r = np.zeros_like(x)
      r[1:] = np.cumsum(dr)                # integrate path
      r_int = np.linspace(0, r.max(), 200) # regular spaced path
      x_int = np.interp(r_int, r, x)       # integrate path
      y_int = np.interp(r_int, r, y)


#. Given an integer n and a 2D array X, select from X the rows which can be
   interpreted as draws from a multinomial distribution with n degrees, i.e.,
   the rows which only contain integers and which sum to n. (★★★)

   .. code-block:: python

      # Author: Evgeni Burovski
      
      X = np.asarray([[1.0, 0.0, 3.0, 8.0],
                      [2.0, 0.0, 1.0, 1.0],
                      [1.5, 2.5, 1.0, 0.0]])
      n = 4
      M = np.logical_and.reduce(np.mod(X, 1) == 0, axis=-1)
      M &= (X.sum(axis=-1) == n)
      print(X[M])

#. Compute bootstrapped 95% confidence intervals for the mean of a 1D array X
   (i.e., resample the elements of an array with replacement N times, compute
   the mean of each sample, and then compute percentiles over the means). (★★★)

   .. code-block:: python

      # Author: Jessica B. Hamrick

      X = np.random.randn(100) # random 1D array
      N = 1000 # number of bootstrap samples
      idx = np.random.randint(0, X.size, (N, X.size))
      means = X[idx].mean(axis=1)
      confint = np.percentile(means, [2.5, 97.5])
      print(confint)