{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Manipulação de matrizes e vetores" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np #chama o pacote cientifico " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['ALLOW_THREADS',\n", " 'AxisError',\n", " 'BUFSIZE',\n", " 'Bytes0',\n", " 'CLIP',\n", " 'ComplexWarning',\n", " 'DataSource',\n", " 'Datetime64',\n", " 'ERR_CALL',\n", " 'ERR_DEFAULT',\n", " 'ERR_IGNORE',\n", " 'ERR_LOG',\n", " 'ERR_PRINT',\n", " 'ERR_RAISE',\n", " 'ERR_WARN',\n", " 'FLOATING_POINT_SUPPORT',\n", " 'FPE_DIVIDEBYZERO',\n", " 'FPE_INVALID',\n", " 'FPE_OVERFLOW',\n", " 'FPE_UNDERFLOW',\n", " 'False_',\n", " 'Inf',\n", " 'Infinity',\n", " 'MAXDIMS',\n", " 'MAY_SHARE_BOUNDS',\n", " 'MAY_SHARE_EXACT',\n", " 'MachAr',\n", " 'ModuleDeprecationWarning',\n", " 'NAN',\n", " 'NINF',\n", " 'NZERO',\n", " 'NaN',\n", " 'PINF',\n", " 'PZERO',\n", " 'RAISE',\n", " 'RankWarning',\n", " 'SHIFT_DIVIDEBYZERO',\n", " 'SHIFT_INVALID',\n", " 'SHIFT_OVERFLOW',\n", " 'SHIFT_UNDERFLOW',\n", " 'ScalarType',\n", " 'Str0',\n", " 'Tester',\n", " 'TooHardError',\n", " 'True_',\n", " 'UFUNC_BUFSIZE_DEFAULT',\n", " 'UFUNC_PYVALS_NAME',\n", " 'Uint64',\n", " 'VisibleDeprecationWarning',\n", " 'WRAP',\n", " '_NoValue',\n", " '_UFUNC_API',\n", " '__NUMPY_SETUP__',\n", " '__all__',\n", " '__builtins__',\n", " '__cached__',\n", " '__config__',\n", " '__deprecated_attrs__',\n", " '__dir__',\n", " '__doc__',\n", " '__expired_functions__',\n", " '__file__',\n", " '__getattr__',\n", " '__git_revision__',\n", " '__loader__',\n", " '__mkl_version__',\n", " '__name__',\n", " '__package__',\n", " '__path__',\n", " '__spec__',\n", " '__version__',\n", " '_add_newdoc_ufunc',\n", " '_distributor_init',\n", " '_financial_names',\n", " '_globals',\n", " '_mat',\n", " '_pytesttester',\n", " 'abs',\n", " 'absolute',\n", " 'add',\n", " 'add_docstring',\n", " 'add_newdoc',\n", " 'add_newdoc_ufunc',\n", " 'alen',\n", " 'all',\n", " 'allclose',\n", " 'alltrue',\n", " 'amax',\n", " 'amin',\n", " 'angle',\n", " 'any',\n", " 'append',\n", " 'apply_along_axis',\n", " 'apply_over_axes',\n", " 'arange',\n", " 'arccos',\n", " 'arccosh',\n", " 'arcsin',\n", " 'arcsinh',\n", " 'arctan',\n", " 'arctan2',\n", " 'arctanh',\n", " 'argmax',\n", " 'argmin',\n", " 'argpartition',\n", " 'argsort',\n", " 'argwhere',\n", " 'around',\n", " 'array',\n", " 'array2string',\n", " 'array_equal',\n", " 'array_equiv',\n", " 'array_repr',\n", " 'array_split',\n", " 'array_str',\n", " 'asanyarray',\n", " 'asarray',\n", " 'asarray_chkfinite',\n", " 'ascontiguousarray',\n", " 'asfarray',\n", " 'asfortranarray',\n", " 'asmatrix',\n", " 'asscalar',\n", " 'atleast_1d',\n", " 'atleast_2d',\n", " 'atleast_3d',\n", " 'average',\n", " 'bartlett',\n", " 'base_repr',\n", " 'binary_repr',\n", " 'bincount',\n", " 'bitwise_and',\n", " 'bitwise_not',\n", " 'bitwise_or',\n", " 'bitwise_xor',\n", " 'blackman',\n", " 'block',\n", " 'bmat',\n", " 'bool8',\n", " 'bool_',\n", " 'broadcast',\n", " 'broadcast_arrays',\n", " 'broadcast_shapes',\n", " 'broadcast_to',\n", " 'busday_count',\n", " 'busday_offset',\n", " 'busdaycalendar',\n", " 'byte',\n", " 'byte_bounds',\n", " 'bytes0',\n", " 'bytes_',\n", " 'c_',\n", " 'can_cast',\n", " 'cast',\n", " 'cbrt',\n", " 'cdouble',\n", " 'ceil',\n", " 'cfloat',\n", " 'char',\n", " 'character',\n", " 'chararray',\n", " 'choose',\n", " 'clip',\n", " 'clongdouble',\n", " 'clongfloat',\n", " 'column_stack',\n", " 'common_type',\n", " 'compare_chararrays',\n", " 'compat',\n", " 'complex128',\n", " 'complex256',\n", " 'complex64',\n", " 'complex_',\n", " 'complexfloating',\n", " 'compress',\n", " 'concatenate',\n", " 'conj',\n", " 'conjugate',\n", " 'convolve',\n", " 'copy',\n", " 'copysign',\n", " 'copyto',\n", " 'core',\n", " 'corrcoef',\n", " 'correlate',\n", " 'cos',\n", " 'cosh',\n", " 'count_nonzero',\n", " 'cov',\n", " 'cross',\n", " 'csingle',\n", " 'ctypeslib',\n", " 'cumprod',\n", " 'cumproduct',\n", " 'cumsum',\n", " 'datetime64',\n", " 'datetime_as_string',\n", " 'datetime_data',\n", " 'deg2rad',\n", " 'degrees',\n", " 'delete',\n", " 'deprecate',\n", " 'deprecate_with_doc',\n", " 'diag',\n", " 'diag_indices',\n", " 'diag_indices_from',\n", " 'diagflat',\n", " 'diagonal',\n", " 'diff',\n", " 'digitize',\n", " 'disp',\n", " 'divide',\n", " 'divmod',\n", " 'dot',\n", " 'double',\n", " 'dsplit',\n", " 'dstack',\n", " 'dtype',\n", " 'e',\n", " 'ediff1d',\n", " 'einsum',\n", " 'einsum_path',\n", " 'emath',\n", " 'empty',\n", " 'empty_like',\n", " 'equal',\n", " 'errstate',\n", " 'euler_gamma',\n", " 'exp',\n", " 'exp2',\n", " 'expand_dims',\n", " 'expm1',\n", " 'extract',\n", " 'eye',\n", " 'fabs',\n", " 'fastCopyAndTranspose',\n", " 'fft',\n", " 'fill_diagonal',\n", " 'find_common_type',\n", " 'finfo',\n", " 'fix',\n", " 'flatiter',\n", " 'flatnonzero',\n", " 'flexible',\n", " 'flip',\n", " 'fliplr',\n", " 'flipud',\n", " 'float128',\n", " 'float16',\n", " 'float32',\n", " 'float64',\n", " 'float_',\n", " 'float_power',\n", " 'floating',\n", " 'floor',\n", " 'floor_divide',\n", " 'fmax',\n", " 'fmin',\n", " 'fmod',\n", " 'format_float_positional',\n", " 'format_float_scientific',\n", " 'format_parser',\n", " 'frexp',\n", " 'frombuffer',\n", " 'fromfile',\n", " 'fromfunction',\n", " 'fromiter',\n", " 'frompyfunc',\n", " 'fromregex',\n", " 'fromstring',\n", " 'full',\n", " 'full_like',\n", " 'gcd',\n", " 'generic',\n", " 'genfromtxt',\n", " 'geomspace',\n", " 'get_array_wrap',\n", " 'get_include',\n", " 'get_printoptions',\n", " 'getbufsize',\n", " 'geterr',\n", " 'geterrcall',\n", " 'geterrobj',\n", " 'gradient',\n", " 'greater',\n", " 'greater_equal',\n", " 'half',\n", " 'hamming',\n", " 'hanning',\n", " 'heaviside',\n", " 'histogram',\n", " 'histogram2d',\n", " 'histogram_bin_edges',\n", " 'histogramdd',\n", " 'hsplit',\n", " 'hstack',\n", " 'hypot',\n", " 'i0',\n", " 'identity',\n", " 'iinfo',\n", " 'imag',\n", " 'in1d',\n", " 'index_exp',\n", " 'indices',\n", " 'inexact',\n", " 'inf',\n", " 'info',\n", " 'infty',\n", " 'inner',\n", " 'insert',\n", " 'int0',\n", " 'int16',\n", " 'int32',\n", " 'int64',\n", " 'int8',\n", " 'int_',\n", " 'intc',\n", " 'integer',\n", " 'interp',\n", " 'intersect1d',\n", " 'intp',\n", " 'invert',\n", " 'is_busday',\n", " 'isclose',\n", " 'iscomplex',\n", " 'iscomplexobj',\n", " 'isfinite',\n", " 'isfortran',\n", " 'isin',\n", " 'isinf',\n", " 'isnan',\n", " 'isnat',\n", " 'isneginf',\n", " 'isposinf',\n", " 'isreal',\n", " 'isrealobj',\n", " 'isscalar',\n", " 'issctype',\n", " 'issubclass_',\n", " 'issubdtype',\n", " 'issubsctype',\n", " 'iterable',\n", " 'ix_',\n", " 'kaiser',\n", " 'kernel_version',\n", " 'kron',\n", " 'lcm',\n", " 'ldexp',\n", " 'left_shift',\n", " 'less',\n", " 'less_equal',\n", " 'lexsort',\n", " 'lib',\n", " 'linalg',\n", " 'linspace',\n", " 'little_endian',\n", " 'load',\n", " 'loads',\n", " 'loadtxt',\n", " 'log',\n", " 'log10',\n", " 'log1p',\n", " 'log2',\n", " 'logaddexp',\n", " 'logaddexp2',\n", " 'logical_and',\n", " 'logical_not',\n", " 'logical_or',\n", " 'logical_xor',\n", " 'logspace',\n", " 'longcomplex',\n", " 'longdouble',\n", " 'longfloat',\n", " 'longlong',\n", " 'lookfor',\n", " 'ma',\n", " 'mafromtxt',\n", " 'mask_indices',\n", " 'mat',\n", " 'math',\n", " 'matmul',\n", " 'matrix',\n", " 'matrixlib',\n", " 'max',\n", " 'maximum',\n", " 'maximum_sctype',\n", " 'may_share_memory',\n", " 'mean',\n", " 'median',\n", " 'memmap',\n", " 'meshgrid',\n", " 'mgrid',\n", " 'min',\n", " 'min_scalar_type',\n", " 'minimum',\n", " 'mintypecode',\n", " 'mkl',\n", " 'mod',\n", " 'modf',\n", " 'moveaxis',\n", " 'msort',\n", " 'multiply',\n", " 'nan',\n", " 'nan_to_num',\n", " 'nanargmax',\n", " 'nanargmin',\n", " 'nancumprod',\n", " 'nancumsum',\n", " 'nanmax',\n", " 'nanmean',\n", " 'nanmedian',\n", " 'nanmin',\n", " 'nanpercentile',\n", " 'nanprod',\n", " 'nanquantile',\n", " 'nanstd',\n", " 'nansum',\n", " 'nanvar',\n", " 'nbytes',\n", " 'ndarray',\n", " 'ndenumerate',\n", " 'ndfromtxt',\n", " 'ndim',\n", " 'ndindex',\n", " 'nditer',\n", " 'negative',\n", " 'nested_iters',\n", " 'newaxis',\n", " 'nextafter',\n", " 'nonzero',\n", " 'not_equal',\n", " 'numarray',\n", " 'number',\n", " 'obj2sctype',\n", " 'object0',\n", " 'object_',\n", " 'ogrid',\n", " 'oldnumeric',\n", " 'ones',\n", " 'ones_like',\n", " 'os',\n", " 'outer',\n", " 'packbits',\n", " 'pad',\n", " 'partition',\n", " 'percentile',\n", " 'pi',\n", " 'piecewise',\n", " 'place',\n", " 'poly',\n", " 'poly1d',\n", " 'polyadd',\n", " 'polyder',\n", " 'polydiv',\n", " 'polyfit',\n", " 'polyint',\n", " 'polymul',\n", " 'polynomial',\n", " 'polysub',\n", " 'polyval',\n", " 'positive',\n", " 'power',\n", " 'printoptions',\n", " 'prod',\n", " 'product',\n", " 'promote_types',\n", " 'ptp',\n", " 'put',\n", " 'put_along_axis',\n", " 'putmask',\n", " 'quantile',\n", " 'r_',\n", " 'rad2deg',\n", " 'radians',\n", " 'random',\n", " 'ravel',\n", " 'ravel_multi_index',\n", " 'real',\n", " 'real_if_close',\n", " 'rec',\n", " 'recarray',\n", " 'recfromcsv',\n", " 'recfromtxt',\n", " 'reciprocal',\n", " 'record',\n", " 'remainder',\n", " 'repeat',\n", " 'require',\n", " 'reshape',\n", " 'resize',\n", " 'result_type',\n", " 'right_shift',\n", " 'rint',\n", " 'roll',\n", " 'rollaxis',\n", " 'roots',\n", " 'rot90',\n", " 'round',\n", " 'round_',\n", " 'row_stack',\n", " 's_',\n", " 'safe_eval',\n", " 'save',\n", " 'savetxt',\n", " 'savez',\n", " 'savez_compressed',\n", " 'sctype2char',\n", " 'sctypeDict',\n", " 'sctypes',\n", " 'searchsorted',\n", " 'select',\n", " 'set_numeric_ops',\n", " 'set_printoptions',\n", " 'set_string_function',\n", " 'setbufsize',\n", " 'setdiff1d',\n", " 'seterr',\n", " 'seterrcall',\n", " 'seterrobj',\n", " 'setxor1d',\n", " 'shape',\n", " 'shares_memory',\n", " 'short',\n", " 'show_config',\n", " 'sign',\n", " 'signbit',\n", " 'signedinteger',\n", " 'sin',\n", " 'sinc',\n", " 'single',\n", " 'singlecomplex',\n", " 'sinh',\n", " 'size',\n", " 'sometrue',\n", " 'sort',\n", " 'sort_complex',\n", " 'source',\n", " 'spacing',\n", " 'split',\n", " 'sqrt',\n", " 'square',\n", " 'squeeze',\n", " 'stack',\n", " 'std',\n", " 'str0',\n", " 'str_',\n", " 'string_',\n", " 'subtract',\n", " 'sum',\n", " 'swapaxes',\n", " 'sys',\n", " 'take',\n", " 'take_along_axis',\n", " 'tan',\n", " 'tanh',\n", " 'tensordot',\n", " 'test',\n", " 'testing',\n", " 'tile',\n", " 'timedelta64',\n", " 'trace',\n", " 'tracemalloc_domain',\n", " 'transpose',\n", " 'trapz',\n", " 'tri',\n", " 'tril',\n", " 'tril_indices',\n", " 'tril_indices_from',\n", " 'trim_zeros',\n", " 'triu',\n", " 'triu_indices',\n", " 'triu_indices_from',\n", " 'true_divide',\n", " 'trunc',\n", " 'typeDict',\n", " 'typecodes',\n", " 'typename',\n", " 'ubyte',\n", " 'ufunc',\n", " 'uint',\n", " 'uint0',\n", " 'uint16',\n", " 'uint32',\n", " 'uint64',\n", " 'uint8',\n", " 'uintc',\n", " 'uintp',\n", " 'ulonglong',\n", " 'unicode_',\n", " 'union1d',\n", " 'unique',\n", " 'unpackbits',\n", " 'unravel_index',\n", " 'unsignedinteger',\n", " 'unwrap',\n", " 'use_hugepage',\n", " 'ushort',\n", " 'vander',\n", " 'var',\n", " 'vdot',\n", " 'vectorize',\n", " 'version',\n", " 'void',\n", " 'void0',\n", " 'vsplit',\n", " 'vstack',\n", " 'warnings',\n", " 'where',\n", " 'who',\n", " 'zeros',\n", " 'zeros_like']" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dir(np)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "A=np.array([[3,4,2],[1,7,6]]) #criando um array bidimensional" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[3, 4, 2],\n", " [1, 7, 6]])" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "B=np.array([[7,8,9],[10,11,5]])" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 7, 8, 9],\n", " [10, 11, 5]])" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "B" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "for(i=0;i(n?,m?) (size 2 is different from 3)", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m/tmp/ipykernel_4919/3859746513.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mC\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m@\u001b[0m\u001b[0mB\u001b[0m \u001b[0;31m#ERRO,o número de coluna de A é diferente de linhas de B\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mValueError\u001b[0m: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 2 is different from 3)" ] } ], "source": [ "C=A@B #ERRO,o número de coluna de A é diferente de linhas de B" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Multiplicação matricial\n", "\n", "#### Amxn Bnp => Cmxp" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "Bt=B.T #B transposto" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 7, 10],\n", " [ 8, 11],\n", " [ 9, 5]])" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Bt" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "D=A@Bt #multiplicaçào de matrizes" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 71, 84],\n", " [117, 117]])" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "D" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "D_=A.dot(Bt) #até o python 2.7" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 71, 84],\n", " [117, 117]])" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "D_" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "ename": "ValueError", "evalue": "shapes (2,3) and (2,3) not aligned: 3 (dim 1) != 2 (dim 0)", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m/tmp/ipykernel_4919/1144695785.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mE\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mB\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mValueError\u001b[0m: shapes (2,3) and (2,3) not aligned: 3 (dim 1) != 2 (dim 0)" ] } ], "source": [ "E=A.dot(B)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Help on built-in function randint:\n", "\n", "randint(...) method of numpy.random.mtrand.RandomState instance\n", " randint(low, high=None, size=None, dtype=int)\n", " \n", " Return random integers from `low` (inclusive) to `high` (exclusive).\n", " \n", " Return random integers from the \"discrete uniform\" distribution of\n", " the specified dtype in the \"half-open\" interval [`low`, `high`). If\n", " `high` is None (the default), then results are from [0, `low`).\n", " \n", " .. note::\n", " New code should use the ``integers`` method of a ``default_rng()``\n", " instance instead; please see the :ref:`random-quick-start`.\n", " \n", " Parameters\n", " ----------\n", " low : int or array-like of ints\n", " Lowest (signed) integers to be drawn from the distribution (unless\n", " ``high=None``, in which case this parameter is one above the\n", " *highest* such integer).\n", " high : int or array-like of ints, optional\n", " If provided, one above the largest (signed) integer to be drawn\n", " from the distribution (see above for behavior if ``high=None``).\n", " If array-like, must contain integer values\n", " size : int or tuple of ints, optional\n", " Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n", " ``m * n * k`` samples are drawn. Default is None, in which case a\n", " single value is returned.\n", " dtype : dtype, optional\n", " Desired dtype of the result. Byteorder must be native.\n", " The default value is int.\n", " \n", " .. versionadded:: 1.11.0\n", " \n", " Returns\n", " -------\n", " out : int or ndarray of ints\n", " `size`-shaped array of random integers from the appropriate\n", " distribution, or a single such random int if `size` not provided.\n", " \n", " See Also\n", " --------\n", " random_integers : similar to `randint`, only for the closed\n", " interval [`low`, `high`], and 1 is the lowest value if `high` is\n", " omitted.\n", " Generator.integers: which should be used for new code.\n", " \n", " Examples\n", " --------\n", " >>> np.random.randint(2, size=10)\n", " array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random\n", " >>> np.random.randint(1, size=10)\n", " array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])\n", " \n", " Generate a 2 x 4 array of ints between 0 and 4, inclusive:\n", " \n", " >>> np.random.randint(5, size=(2, 4))\n", " array([[4, 0, 2, 1], # random\n", " [3, 2, 2, 0]])\n", " \n", " Generate a 1 x 3 array with 3 different upper bounds\n", " \n", " >>> np.random.randint(1, [3, 5, 10])\n", " array([2, 2, 9]) # random\n", " \n", " Generate a 1 by 3 array with 3 different lower bounds\n", " \n", " >>> np.random.randint([1, 5, 7], 10)\n", " array([9, 8, 7]) # random\n", " \n", " Generate a 2 by 4 array using broadcasting with dtype of uint8\n", " \n", " >>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)\n", " array([[ 8, 6, 9, 7], # random\n", " [ 1, 16, 9, 12]], dtype=uint8)\n", "\n" ] } ], "source": [ "help(np.random.randint)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "E=np.random.randint(100,size=(10,2))" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[52, 86],\n", " [57, 41],\n", " [78, 50],\n", " [62, 75],\n", " [ 3, 26],\n", " [69, 41],\n", " [58, 78],\n", " [ 9, 80],\n", " [81, 44],\n", " [96, 2]])" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "V=np.random.randint(50,size=3)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([24, 39, 40])" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [], "source": [ "U=np.random.randint(50,size=3)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([11, 10, 49])" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [], "source": [ "R=U.T@V " ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2614" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [], "source": [ "k=U.dot(V)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2614" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['T',\n", " '__abs__',\n", " '__add__',\n", " '__and__',\n", " '__array__',\n", " '__array_finalize__',\n", " '__array_function__',\n", " '__array_interface__',\n", " '__array_prepare__',\n", " '__array_priority__',\n", " '__array_struct__',\n", " '__array_ufunc__',\n", " '__array_wrap__',\n", " '__bool__',\n", " '__class__',\n", " '__complex__',\n", " '__contains__',\n", " '__copy__',\n", " '__deepcopy__',\n", " '__delattr__',\n", " '__delitem__',\n", " '__dir__',\n", " '__divmod__',\n", " '__doc__',\n", " '__eq__',\n", " '__float__',\n", " '__floordiv__',\n", " '__format__',\n", " '__ge__',\n", " '__getattribute__',\n", " '__getitem__',\n", " '__gt__',\n", " '__hash__',\n", " '__iadd__',\n", " '__iand__',\n", " '__ifloordiv__',\n", " '__ilshift__',\n", " '__imatmul__',\n", " '__imod__',\n", " '__imul__',\n", " '__index__',\n", " '__init__',\n", " '__init_subclass__',\n", " '__int__',\n", " '__invert__',\n", " '__ior__',\n", " '__ipow__',\n", " '__irshift__',\n", " '__isub__',\n", " '__iter__',\n", " '__itruediv__',\n", " '__ixor__',\n", " '__le__',\n", " '__len__',\n", " '__lshift__',\n", " '__lt__',\n", " '__matmul__',\n", " '__mod__',\n", " '__mul__',\n", " '__ne__',\n", " '__neg__',\n", " '__new__',\n", " '__or__',\n", " '__pos__',\n", " '__pow__',\n", " '__radd__',\n", " '__rand__',\n", " '__rdivmod__',\n", " '__reduce__',\n", " '__reduce_ex__',\n", " '__repr__',\n", " '__rfloordiv__',\n", " '__rlshift__',\n", " '__rmatmul__',\n", " '__rmod__',\n", " '__rmul__',\n", " '__ror__',\n", " '__rpow__',\n", " '__rrshift__',\n", " '__rshift__',\n", " '__rsub__',\n", " '__rtruediv__',\n", " '__rxor__',\n", " '__setattr__',\n", " '__setitem__',\n", " '__setstate__',\n", " '__sizeof__',\n", " '__str__',\n", " '__sub__',\n", " '__subclasshook__',\n", " '__truediv__',\n", " '__xor__',\n", " 'all',\n", " 'any',\n", " 'argmax',\n", " 'argmin',\n", " 'argpartition',\n", " 'argsort',\n", " 'astype',\n", " 'base',\n", " 'byteswap',\n", " 'choose',\n", " 'clip',\n", " 'compress',\n", " 'conj',\n", " 'conjugate',\n", " 'copy',\n", " 'ctypes',\n", " 'cumprod',\n", " 'cumsum',\n", " 'data',\n", " 'diagonal',\n", " 'dot',\n", " 'dtype',\n", " 'dump',\n", " 'dumps',\n", " 'fill',\n", " 'flags',\n", " 'flat',\n", " 'flatten',\n", " 'getfield',\n", " 'imag',\n", " 'item',\n", " 'itemset',\n", " 'itemsize',\n", " 'max',\n", " 'mean',\n", " 'min',\n", " 'nbytes',\n", " 'ndim',\n", " 'newbyteorder',\n", " 'nonzero',\n", " 'partition',\n", " 'prod',\n", " 'ptp',\n", " 'put',\n", " 'ravel',\n", " 'real',\n", " 'repeat',\n", " 'reshape',\n", " 'resize',\n", " 'round',\n", " 'searchsorted',\n", " 'setfield',\n", " 'setflags',\n", " 'shape',\n", " 'size',\n", " 'sort',\n", " 'squeeze',\n", " 'std',\n", " 'strides',\n", " 'sum',\n", " 'swapaxes',\n", " 'take',\n", " 'tobytes',\n", " 'tofile',\n", " 'tolist',\n", " 'tostring',\n", " 'trace',\n", " 'transpose',\n", " 'var',\n", " 'view']" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dir(U)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [], "source": [ "u=U.reshape(1,3)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[11, 10, 49]])" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[24],\n", " [39],\n", " [40]])" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v=V.reshape(3,1)\n", "v" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [], "source": [ "k=u@v" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(1, 1)" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.shape" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(3, 1)" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.shape" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [], "source": [ "U=np.random.randint(50,size=(3,1))" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[29],\n", " [24],\n", " [10]])" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "u=np.random.randint(50,size=(3,1))" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[34],\n", " [ 9],\n", " [23]])" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [], "source": [ "v=np.random.randint(50,size=(3,1))" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 5],\n", " [26],\n", " [15]])" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[170, 884, 510],\n", " [ 45, 234, 135],\n", " [115, 598, 345]])" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u@v.T" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 5],\n", " [26],\n", " [15]])" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[34],\n", " [ 9],\n", " [23]])" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(1, 3)" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Um=np.matrix([11, 10, 49])\n", "Um.shape" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(1, 3)" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Vm=np.matrix([24, 39, 40])\n", "Vm.shape" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "matrix([[2614]])" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Um@Vm.T" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([24, 39, 40])" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Um=np.array([24, 39, 40])\n", "Um" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[749]])" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.T@v" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [], "source": [ "B=np.array([[7,8,9],[10,11,5]], dtype=float)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 7., 8., 9.],\n", " [10., 11., 5.]])" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "B" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "B=np.array([[7,8,9],[10,11,5]],dtype=\"float\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "B" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [], "source": [ "A=np.array([[1,2,3],[7,8,9],[4,5,6]])" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[1, 2, 3],\n", " [7, 8, 9],\n", " [4, 5, 6]])" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ "import scipy.linalg as al #importando o modulo linalg" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['LinAlgError',\n", " 'LinAlgWarning',\n", " '__all__',\n", " '__builtins__',\n", " '__cached__',\n", " '__doc__',\n", " '__file__',\n", " '__loader__',\n", " '__name__',\n", " '__package__',\n", " '__path__',\n", " '__spec__',\n", " '_decomp_cossin',\n", " '_decomp_ldl',\n", " '_decomp_polar',\n", " '_decomp_qz',\n", " '_decomp_update',\n", " '_expm_frechet',\n", " '_fblas',\n", " '_flapack',\n", " '_flinalg',\n", " '_matfuncs_sqrtm',\n", " '_matfuncs_sqrtm_triu',\n", " '_procrustes',\n", " '_sketches',\n", " '_solve_toeplitz',\n", " '_solvers',\n", " 'basic',\n", " 'blas',\n", " 'block_diag',\n", " 'cdf2rdf',\n", " 'cho_factor',\n", " 'cho_solve',\n", " 'cho_solve_banded',\n", " 'cholesky',\n", " 'cholesky_banded',\n", " 'circulant',\n", " 'clarkson_woodruff_transform',\n", " 'companion',\n", " 'convolution_matrix',\n", " 'coshm',\n", " 'cosm',\n", " 'cossin',\n", " 'cython_blas',\n", " 'cython_lapack',\n", " 'decomp',\n", " 'decomp_cholesky',\n", " 'decomp_lu',\n", " 'decomp_qr',\n", " 'decomp_schur',\n", " 'decomp_svd',\n", " 'det',\n", " 'dft',\n", " 'diagsvd',\n", " 'eig',\n", " 'eig_banded',\n", " 'eigh',\n", " 'eigh_tridiagonal',\n", " 'eigvals',\n", " 'eigvals_banded',\n", " 'eigvalsh',\n", " 'eigvalsh_tridiagonal',\n", " 'expm',\n", " 'expm_cond',\n", " 'expm_frechet',\n", " 'fiedler',\n", " 'fiedler_companion',\n", " 'find_best_blas_type',\n", " 'flinalg',\n", " 'fractional_matrix_power',\n", " 'funm',\n", " 'get_blas_funcs',\n", " 'get_lapack_funcs',\n", " 'hadamard',\n", " 'hankel',\n", " 'helmert',\n", " 'hessenberg',\n", " 'hilbert',\n", " 'inv',\n", " 'invhilbert',\n", " 'invpascal',\n", " 'khatri_rao',\n", " 'kron',\n", " 'lapack',\n", " 'ldl',\n", " 'leslie',\n", " 'logm',\n", " 'lstsq',\n", " 'lu',\n", " 'lu_factor',\n", " 'lu_solve',\n", " 'matfuncs',\n", " 'matmul_toeplitz',\n", " 'matrix_balance',\n", " 'misc',\n", " 'norm',\n", " 'null_space',\n", " 'ordqz',\n", " 'orth',\n", " 'orthogonal_procrustes',\n", " 'pascal',\n", " 'pinv',\n", " 'pinv2',\n", " 'pinvh',\n", " 'polar',\n", " 'qr',\n", " 'qr_delete',\n", " 'qr_insert',\n", " 'qr_multiply',\n", " 'qr_update',\n", " 'qz',\n", " 'rq',\n", " 'rsf2csf',\n", " 'schur',\n", " 'signm',\n", " 'sinhm',\n", " 'sinm',\n", " 'solve',\n", " 'solve_banded',\n", " 'solve_circulant',\n", " 'solve_continuous_are',\n", " 'solve_continuous_lyapunov',\n", " 'solve_discrete_are',\n", " 'solve_discrete_lyapunov',\n", " 'solve_lyapunov',\n", " 'solve_sylvester',\n", " 'solve_toeplitz',\n", " 'solve_triangular',\n", " 'solveh_banded',\n", " 'special_matrices',\n", " 'sqrtm',\n", " 'subspace_angles',\n", " 'svd',\n", " 'svdvals',\n", " 'tanhm',\n", " 'tanm',\n", " 'test',\n", " 'toeplitz',\n", " 'tri',\n", " 'tril',\n", " 'triu']" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dir(al) #verificando os metodos do modulo linalg" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(array([ 1.55777472e+01+0.j, -5.77747211e-01+0.j, -4.40162414e-17+0.j]),\n", " array([[ 0.22048676, 0.75018389, 0.40824829],\n", " [ 0.82384859, -0.65967897, -0.81649658],\n", " [ 0.52216768, 0.04525246, 0.40824829]]))" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "al.eig(A) #a : autovalor, v: autovetor A.v = a.v, eig retorna D: conjunto de autovalores em matriz diagonal e V: conjunto de autovetores, tal que : AV = VD" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [], "source": [ "A=np.array([[1,-1],[1,-2]]) # |1 - 1 | |x| |x-y |\n", " # | |*| |=| |\n", " # |1 - 2 | |y| |x-2y|" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 1, -1],\n", " [ 1, -2]])" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [], "source": [ "b=np.array([[1],[-2]]) # |1 - 1 | |x| |1 | |x| |1 | |1 - 1 | -1\n", " # | |*| |=| | ==> | |= | |/ | | AX = b ==> X = A b\n", " # |1 - 2 | |y| |-2| |y| |-2| |1 - 2 | " ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 1],\n", " [-2]])" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['LinAlgError',\n", " 'LinAlgWarning',\n", " '__all__',\n", " '__builtins__',\n", " '__cached__',\n", " '__doc__',\n", " '__file__',\n", " '__loader__',\n", " '__name__',\n", " '__package__',\n", " '__path__',\n", " '__spec__',\n", " '_decomp_cossin',\n", " '_decomp_ldl',\n", " '_decomp_polar',\n", " '_decomp_qz',\n", " '_decomp_update',\n", " '_expm_frechet',\n", " '_fblas',\n", " '_flapack',\n", " '_flinalg',\n", " '_matfuncs_sqrtm',\n", " '_procrustes',\n", " '_sketches',\n", " '_solve_toeplitz',\n", " '_solvers',\n", " 'basic',\n", " 'blas',\n", " 'block_diag',\n", " 'cdf2rdf',\n", " 'cho_factor',\n", " 'cho_solve',\n", " 'cho_solve_banded',\n", " 'cholesky',\n", " 'cholesky_banded',\n", " 'circulant',\n", " 'clarkson_woodruff_transform',\n", " 'companion',\n", " 'convolution_matrix',\n", " 'coshm',\n", " 'cosm',\n", " 'cossin',\n", " 'cython_blas',\n", " 'cython_lapack',\n", " 'decomp',\n", " 'decomp_cholesky',\n", " 'decomp_lu',\n", " 'decomp_qr',\n", " 'decomp_schur',\n", " 'decomp_svd',\n", " 'det',\n", " 'dft',\n", " 'diagsvd',\n", " 'eig',\n", " 'eig_banded',\n", " 'eigh',\n", " 'eigh_tridiagonal',\n", " 'eigvals',\n", " 'eigvals_banded',\n", " 'eigvalsh',\n", " 'eigvalsh_tridiagonal',\n", " 'expm',\n", " 'expm_cond',\n", " 'expm_frechet',\n", " 'fiedler',\n", " 'fiedler_companion',\n", " 'find_best_blas_type',\n", " 'flinalg',\n", " 'fractional_matrix_power',\n", " 'funm',\n", " 'get_blas_funcs',\n", " 'get_lapack_funcs',\n", " 'hadamard',\n", " 'hankel',\n", " 'helmert',\n", " 'hessenberg',\n", " 'hilbert',\n", " 'inv',\n", " 'invhilbert',\n", " 'invpascal',\n", " 'khatri_rao',\n", " 'kron',\n", " 'lapack',\n", " 'ldl',\n", " 'leslie',\n", " 'logm',\n", " 'lstsq',\n", " 'lu',\n", " 'lu_factor',\n", " 'lu_solve',\n", " 'matfuncs',\n", " 'matrix_balance',\n", " 'misc',\n", " 'norm',\n", " 'null_space',\n", " 'ordqz',\n", " 'orth',\n", " 'orthogonal_procrustes',\n", " 'pascal',\n", " 'pinv',\n", " 'pinv2',\n", " 'pinvh',\n", " 'polar',\n", " 'qr',\n", " 'qr_delete',\n", " 'qr_insert',\n", " 'qr_multiply',\n", " 'qr_update',\n", " 'qz',\n", " 'rq',\n", " 'rsf2csf',\n", " 'schur',\n", " 'signm',\n", " 'sinhm',\n", " 'sinm',\n", " 'solve',\n", " 'solve_banded',\n", " 'solve_circulant',\n", " 'solve_continuous_are',\n", " 'solve_continuous_lyapunov',\n", " 'solve_discrete_are',\n", " 'solve_discrete_lyapunov',\n", " 'solve_lyapunov',\n", " 'solve_sylvester',\n", " 'solve_toeplitz',\n", " 'solve_triangular',\n", " 'solveh_banded',\n", " 'special_matrices',\n", " 'sqrtm',\n", " 'subspace_angles',\n", " 'svd',\n", " 'svdvals',\n", " 'tanhm',\n", " 'tanm',\n", " 'test',\n", " 'toeplitz',\n", " 'tri',\n", " 'tril',\n", " 'triu']" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dir(al)" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 1, -1],\n", " [ 1, -2]])" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 1],\n", " [-2]])" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, 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