Lecture Notes 11#
dir()
['In',
'Out',
'_',
'__',
'___',
'__builtin__',
'__builtins__',
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dir(__builtins__)
['ArithmeticError',
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'True',
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'aiter',
'all',
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'any',
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'chr',
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'compile',
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'copyright',
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'dir',
'display',
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'enumerate',
'eval',
'exec',
'execfile',
'filter',
'float',
'format',
'frozenset',
'get_ipython',
'getattr',
'globals',
'hasattr',
'hash',
'help',
'hex',
'id',
'input',
'int',
'isinstance',
'issubclass',
'iter',
'len',
'license',
'list',
'locals',
'map',
'max',
'memoryview',
'min',
'next',
'object',
'oct',
'open',
'ord',
'pow',
'print',
'property',
'range',
'repr',
'reversed',
'round',
'runfile',
'set',
'setattr',
'slice',
'sorted',
'staticmethod',
'str',
'sum',
'super',
'tuple',
'type',
'vars',
'zip']
import numpy
dir(numpy)
['ALLOW_THREADS',
'AxisError',
'BUFSIZE',
'CLIP',
'ComplexWarning',
'DataSource',
'ERR_CALL',
'ERR_DEFAULT',
'ERR_IGNORE',
'ERR_LOG',
'ERR_PRINT',
'ERR_RAISE',
'ERR_WARN',
'FLOATING_POINT_SUPPORT',
'FPE_DIVIDEBYZERO',
'FPE_INVALID',
'FPE_OVERFLOW',
'FPE_UNDERFLOW',
'False_',
'Inf',
'Infinity',
'MAXDIMS',
'MAY_SHARE_BOUNDS',
'MAY_SHARE_EXACT',
'ModuleDeprecationWarning',
'NAN',
'NINF',
'NZERO',
'NaN',
'PINF',
'PZERO',
'RAISE',
'RankWarning',
'SHIFT_DIVIDEBYZERO',
'SHIFT_INVALID',
'SHIFT_OVERFLOW',
'SHIFT_UNDERFLOW',
'ScalarType',
'Tester',
'TooHardError',
'True_',
'UFUNC_BUFSIZE_DEFAULT',
'UFUNC_PYVALS_NAME',
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'WRAP',
'_CopyMode',
'_NoValue',
'_UFUNC_API',
'__NUMPY_SETUP__',
'__all__',
'__builtins__',
'__cached__',
'__config__',
'__deprecated_attrs__',
'__dir__',
'__doc__',
'__expired_functions__',
'__file__',
'__getattr__',
'__git_version__',
'__loader__',
'__mkl_version__',
'__name__',
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'__path__',
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'arccos',
'arccosh',
'arcsin',
'arcsinh',
'arctan',
'arctan2',
'arctanh',
'argmax',
'argmin',
'argpartition',
'argsort',
'argwhere',
'around',
'array',
'array2string',
'array_equal',
'array_equiv',
'array_repr',
'array_split',
'array_str',
'asanyarray',
'asarray',
'asarray_chkfinite',
'ascontiguousarray',
'asfarray',
'asfortranarray',
'asmatrix',
'atleast_1d',
'atleast_2d',
'atleast_3d',
'average',
'bartlett',
'base_repr',
'binary_repr',
'bincount',
'bitwise_and',
'bitwise_not',
'bitwise_or',
'bitwise_xor',
'blackman',
'block',
'bmat',
'bool8',
'bool_',
'broadcast',
'broadcast_arrays',
'broadcast_shapes',
'broadcast_to',
'busday_count',
'busday_offset',
'busdaycalendar',
'byte',
'byte_bounds',
'bytes0',
'bytes_',
'c_',
'can_cast',
'cast',
'cbrt',
'cdouble',
'ceil',
'cfloat',
'char',
'character',
'chararray',
'choose',
'clip',
'clongdouble',
'clongfloat',
'column_stack',
'common_type',
'compare_chararrays',
'compat',
'complex128',
'complex256',
'complex64',
'complex_',
'complexfloating',
'compress',
'concatenate',
'conj',
'conjugate',
'convolve',
'copy',
'copysign',
'copyto',
'core',
'corrcoef',
'correlate',
'cos',
'cosh',
'count_nonzero',
'cov',
'cross',
'csingle',
'ctypeslib',
'cumprod',
'cumproduct',
'cumsum',
'datetime64',
'datetime_as_string',
'datetime_data',
'deg2rad',
'degrees',
'delete',
'deprecate',
'deprecate_with_doc',
'diag',
'diag_indices',
'diag_indices_from',
'diagflat',
'diagonal',
'diff',
'digitize',
'disp',
'divide',
'divmod',
'dot',
'double',
'dsplit',
'dstack',
'dtype',
'e',
'ediff1d',
'einsum',
'einsum_path',
'emath',
'empty',
'empty_like',
'equal',
'errstate',
'euler_gamma',
'exp',
'exp2',
'expand_dims',
'expm1',
'extract',
'eye',
'fabs',
'fastCopyAndTranspose',
'fft',
'fill_diagonal',
'find_common_type',
'finfo',
'fix',
'flatiter',
'flatnonzero',
'flexible',
'flip',
'fliplr',
'flipud',
'float128',
'float16',
'float32',
'float64',
'float_',
'float_power',
'floating',
'floor',
'floor_divide',
'fmax',
'fmin',
'fmod',
'format_float_positional',
'format_float_scientific',
'format_parser',
'frexp',
'from_dlpack',
'frombuffer',
'fromfile',
'fromfunction',
'fromiter',
'frompyfunc',
'fromregex',
'fromstring',
'full',
'full_like',
'gcd',
'generic',
'genfromtxt',
'geomspace',
'get_array_wrap',
'get_include',
'get_printoptions',
'getbufsize',
'geterr',
'geterrcall',
'geterrobj',
'gradient',
'greater',
'greater_equal',
'half',
'hamming',
'hanning',
'heaviside',
'histogram',
'histogram2d',
'histogram_bin_edges',
'histogramdd',
'hsplit',
'hstack',
'hypot',
'i0',
'identity',
'iinfo',
'imag',
'in1d',
'index_exp',
'indices',
'inexact',
'inf',
'info',
'infty',
'inner',
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'int0',
'int16',
'int32',
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'int8',
'int_',
'intc',
'integer',
'interp',
'intersect1d',
'intp',
'invert',
'is_busday',
'isclose',
'iscomplex',
'iscomplexobj',
'isfinite',
'isfortran',
'isin',
'isinf',
'isnan',
'isnat',
'isneginf',
'isposinf',
'isreal',
'isrealobj',
'isscalar',
'issctype',
'issubclass_',
'issubdtype',
'issubsctype',
'iterable',
'ix_',
'kaiser',
'kernel_version',
'kron',
'lcm',
'ldexp',
'left_shift',
'less',
'less_equal',
'lexsort',
'lib',
'linalg',
'linspace',
'little_endian',
'load',
'loadtxt',
'log',
'log10',
'log1p',
'log2',
'logaddexp',
'logaddexp2',
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'logical_xor',
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'longdouble',
'longfloat',
'longlong',
'lookfor',
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'mask_indices',
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'matmul',
'matrix',
'matrixlib',
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'minimum',
'mintypecode',
'mkl',
'mod',
'modf',
'moveaxis',
'msort',
'multiply',
'nan',
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'nanargmax',
'nanargmin',
'nancumprod',
'nancumsum',
'nanmax',
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'nanmedian',
'nanmin',
'nanpercentile',
'nanprod',
'nanquantile',
'nanstd',
'nansum',
'nanvar',
'nbytes',
'ndarray',
'ndenumerate',
'ndim',
'ndindex',
'nditer',
'negative',
'nested_iters',
'newaxis',
'nextafter',
'nonzero',
'not_equal',
'numarray',
'number',
'obj2sctype',
'object0',
'object_',
'ogrid',
'oldnumeric',
'ones',
'ones_like',
'os',
'outer',
'packbits',
'pad',
'partition',
'percentile',
'pi',
'piecewise',
'place',
'poly',
'poly1d',
'polyadd',
'polyder',
'polydiv',
'polyfit',
'polyint',
'polymul',
'polynomial',
'polysub',
'polyval',
'positive',
'power',
'printoptions',
'prod',
'product',
'promote_types',
'ptp',
'put',
'put_along_axis',
'putmask',
'quantile',
'r_',
'rad2deg',
'radians',
'random',
'ravel',
'ravel_multi_index',
'real',
'real_if_close',
'rec',
'recarray',
'recfromcsv',
'recfromtxt',
'reciprocal',
'record',
'remainder',
'repeat',
'require',
'reshape',
'resize',
'result_type',
'right_shift',
'rint',
'roll',
'rollaxis',
'roots',
'rot90',
'round',
'round_',
'row_stack',
's_',
'safe_eval',
'save',
'savetxt',
'savez',
'savez_compressed',
'sctype2char',
'sctypeDict',
'sctypes',
'searchsorted',
'select',
'set_numeric_ops',
'set_printoptions',
'set_string_function',
'setbufsize',
'setdiff1d',
'seterr',
'seterrcall',
'seterrobj',
'setxor1d',
'shape',
'shares_memory',
'short',
'show_config',
'sign',
'signbit',
'signedinteger',
'sin',
'sinc',
'single',
'singlecomplex',
'sinh',
'size',
'sometrue',
'sort',
'sort_complex',
'source',
'spacing',
'split',
'sqrt',
'square',
'squeeze',
'stack',
'std',
'str0',
'str_',
'string_',
'subtract',
'sum',
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'sys',
'take',
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'tan',
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'tensordot',
'test',
'testing',
'tile',
'timedelta64',
'trace',
'tracemalloc_domain',
'transpose',
'trapz',
'tri',
'tril',
'tril_indices',
'tril_indices_from',
'trim_zeros',
'triu',
'triu_indices',
'triu_indices_from',
'true_divide',
'trunc',
'typecodes',
'typename',
'ubyte',
'ufunc',
'uint',
'uint0',
'uint16',
'uint32',
'uint64',
'uint8',
'uintc',
'uintp',
'ulonglong',
'unicode_',
'union1d',
'unique',
'unpackbits',
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'unsignedinteger',
'unwrap',
'use_hugepage',
'ushort',
'vander',
'var',
'vdot',
'vectorize',
'version',
'void',
'void0',
'vsplit',
'vstack',
'warnings',
'where',
'who',
'zeros',
'zeros_like']
Example#
Put right-hand side and coefficients into numpy arrays
rhs = numpy.array([30, 18, 2])
many operations such as indexing, slicing as for lists
rhs[0]
30
rhs[1]
18
rhs[-1]
2
The dimensions of the arrays is in the
shapeattribute
rhs.shape
(3,)
coef = numpy.array(
[
[3, 0, 0],
[1, 8, 0],
[0, 4, -2]
]
)
coef
array([[ 3, 0, 0],
[ 1, 8, 0],
[ 0, 4, -2]])
coef.shape
(3, 3)
solution = numpy.linalg.solve(coef, rhs)
solution
array([10., 1., 1.])
Given the solution the starting equation involves a matrix multiplication
\[\begin{split}
\begin{pmatrix}3 & 0 &0\\1&8&0\\0&4&-2\end{pmatrix}
\begin{pmatrix}10\\1\\1\end{pmatrix} =
\begin{pmatrix}30\\18\\2\end{pmatrix}
\end{split}\]
coef @ solution
array([30., 18., 2.])
rhs
array([30, 18, 2])
Other notations for the multiplication: dot
numpy.dot(solution, numpy.array([1, 3, 1]))
14.0
The value of the last expression
numpy.dot(solution, [1, 3, 1])
14.0
solution @ [1, 3, 1]
14.0
Generating arrays with sequences of values
list(range(1, 10, 2))
[1, 3, 5, 7, 9]
numpy.array(range(1, 10, 2))
array([1, 3, 5, 7, 9])
numpy.arange(1, 10, 2)
array([1, 3, 5, 7, 9])
numpy.arange(1, 10, .1)b
Cell In[22], line 1
numpy.arange(1, 10, .1)b
^
SyntaxError: invalid syntax
Reading file data to an array: loadtxt#
%%file a.dat
1 2 3
4 5 6
Writing a.dat
Doing this manually … need to convert string to numbers and looping
numlines = []
with open('a.dat') as f:
for line in f:
numline = [int(x) for x in line.split()]
numlines.append(numline)
numpy.array(numlines)
array([[1, 2, 3],
[4, 5, 6]])
or perhaps with less intuitive construct
numpy.array(
[[int(x) for x in line.split()] for line in open('a.dat')]
)
array([[1, 2, 3],
[4, 5, 6]])
There is an efficient numpy function for this: loadtxt
a = numpy.loadtxt('a.dat')
a
array([[1., 2., 3.],
[4., 5., 6.]])
a.shape
(2, 3)
Numpy arrays have a transpose
a.T
array([[1., 4.],
[2., 5.],
[3., 6.]])
a.T.shape
(3, 2)
slicing:
a
array([[1., 2., 3.],
[4., 5., 6.]])
a[0, 0] #upper left
1.0
a[0]
array([1., 2., 3.])
a[:, :] # all elements
array([[1., 2., 3.],
[4., 5., 6.]])
a[:, :2] # first two columns , all rows
array([[1., 2.],
[4., 5.]])
a[-1, :] # last row, all columns
array([4., 5., 6.])
Matrix multiplication: brute force implementation of mathematical definition
\[C_{ij} = \sum_{k}A_{ik}B_{kj}\]
slow
Python is not meant for this
import time
import numpy
n = 256
a = numpy.ones((n, n))
b = numpy.ones((n, n))
c = numpy.zeros((n, n))
t1 = time.time()
for i in range(n):
for j in range(n):
for k in range(n):
c[i, j] += a[i, k]*b[k, j]
t2 = time.time()
print("Loop timing", t2-t1)
Loop timing 9.18215012550354
t1 = time.time()
c = a @ b
t2 = time.time()
print("dot timing", t2-t1)
dot timing 0.012001276016235352
Arrays in mathematical expressions#
a = numpy.arange(10)
a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
a + a
array([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])
a - a
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
a * 2
array([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])
a * a
array([ 0, 1, 4, 9, 16, 25, 36, 49, 64, 81])
a ** 2
array([ 0, 1, 4, 9, 16, 25, 36, 49, 64, 81])
a @ a
285
sum(a * a)
285
a / a
/tmp/ipykernel_297056/3063000341.py:1: RuntimeWarning: invalid value encountered in divide
a / a
array([nan, 1., 1., 1., 1., 1., 1., 1., 1., 1.])
numpy.log(a)
/tmp/ipykernel_297056/4046179305.py:1: RuntimeWarning: divide by zero encountered in log
numpy.log(a)
array([ -inf, 0. , 0.69314718, 1.09861229, 1.38629436,
1.60943791, 1.79175947, 1.94591015, 2.07944154, 2.19722458])
plotting#
import matplotlib.pyplot as plt
plt.plot(a)
plt.plot(a**2)
[<matplotlib.lines.Line2D at 0x7efcdfb0d950>]
plt.plot(numpy.sin(a))
[<matplotlib.lines.Line2D at 0x7efcdf984b10>]
