# TODO: interpret list as vectors (for multiplication)
# pickling helper
def _make_matrix(x):
from mpmath import mp
return mp.matrix(x)
class _matrix:
"""
Numerical matrix.
Specify the dimensions or the data as a nested list.
Elements default to zero.
Use a flat list to create a column vector easily.
The datatype of the context (mpf for mp, mpi for iv, and float for fp) is used to store the data.
Creating matrices
-----------------
Matrices in mpmath are implemented using dictionaries. Only non-zero values
are stored, so it is cheap to represent sparse matrices.
The most basic way to create one is to use the ``matrix`` class directly.
You can create an empty matrix specifying the dimensions:
>>> from mpmath import (mp, matrix, randmatrix, nprint, ones, norm,
... mnorm, inf)
>>> mp.dps = 15
>>> matrix(2)
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> matrix(2, 3)
matrix(
[['0.0', '0.0', '0.0'],
['0.0', '0.0', '0.0']])
Calling ``matrix`` with one dimension will create a square matrix.
To access the dimensions of a matrix, use the ``rows`` or ``cols`` keyword:
>>> A = matrix(3, 2)
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0'],
['0.0', '0.0']])
>>> A.rows
3
>>> A.cols
2
You can also change the dimension of an existing matrix. This will set the
new elements to 0. If the new dimension is smaller than before, the
concerning elements are discarded:
>>> A.rows = 2
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
Internally ``mpmathify`` is used every time an element is set. This
is done using the syntax A[row,column], counting from 0:
>>> A = matrix(2)
>>> A[1,1] = 1 + 1j
>>> A
matrix(
[['0.0', '0.0'],
['0.0', mpc(real='1.0', imag='1.0')]])
A more comfortable way to create a matrix lets you use nested lists:
>>> matrix([[1, 2], [3, 4]])
matrix(
[['1.0', '2.0'],
['3.0', '4.0']])
Convenient advanced functions are available for creating various standard
matrices, see ``zeros``, ``ones``, ``diag``, ``eye``, ``randmatrix`` and
``hilbert``.
Vectors
.......
Vectors may also be represented by the ``matrix`` class (with rows = 1 or cols = 1).
For vectors there are some things which make life easier. A column vector can
be created using a flat list, a row vectors using an almost flat nested list::
>>> matrix([1, 2, 3])
matrix(
[['1.0'],
['2.0'],
['3.0']])
>>> matrix([[1, 2, 3]])
matrix(
[['1.0', '2.0', '3.0']])
Optionally vectors can be accessed like lists, using only a single index::
>>> x = matrix([1, 2, 3])
>>> x[1]
mpf('2.0')
>>> x[1,0]
mpf('2.0')
It is also possible to access matrices and vectors via negative index::
>>> x = matrix([[1, 2], [3, 4]])
>>> y = matrix([6, 7])
>>> x[-1, -2]
mpf('3.0')
>>> y[-2]
mpf('6.0')
Other
.....
Like you probably expected, matrices can be printed::
>>> print randmatrix(3) # doctest:+SKIP
[ 0.782963853573023 0.802057689719883 0.427895717335467]
[0.0541876859348597 0.708243266653103 0.615134039977379]
[ 0.856151514955773 0.544759264818486 0.686210904770947]
Use ``nstr`` or ``nprint`` to specify the number of digits to print::
>>> nprint(randmatrix(5), 3) # doctest:+SKIP
[2.07e-1 1.66e-1 5.06e-1 1.89e-1 8.29e-1]
[6.62e-1 6.55e-1 4.47e-1 4.82e-1 2.06e-2]
[4.33e-1 7.75e-1 6.93e-2 2.86e-1 5.71e-1]
[1.01e-1 2.53e-1 6.13e-1 3.32e-1 2.59e-1]
[1.56e-1 7.27e-2 6.05e-1 6.67e-2 2.79e-1]
As matrices are mutable, you will need to copy them sometimes::
>>> A = matrix(2)
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> B = A.copy()
>>> B[0,0] = 1
>>> B
matrix(
[['1.0', '0.0'],
['0.0', '0.0']])
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
Finally, it is possible to convert a matrix to a nested list. This is very useful,
as most Python libraries involving matrices or arrays (namely NumPy or SymPy)
support this format::
>>> B.tolist()
[[mpf('1.0'), mpf('0.0')], [mpf('0.0'), mpf('0.0')]]
Matrix operations
-----------------
You can add and subtract matrices of compatible dimensions::
>>> A = matrix([[1, 2], [3, 4]])
>>> B = matrix([[-2, 4], [5, 9]])
>>> A + B
matrix(
[['-1.0', '6.0'],
['8.0', '13.0']])
>>> A - B
matrix(
[['3.0', '-2.0'],
['-2.0', '-5.0']])
>>> A + ones(3)
Traceback (most recent call last):
...
ValueError: incompatible dimensions for addition
It is possible to multiply or add matrices and scalars. In the latter case the
operation will be done element-wise::
>>> A * 2
matrix(
[['2.0', '4.0'],
['6.0', '8.0']])
>>> A / 4
matrix(
[['0.25', '0.5'],
['0.75', '1.0']])
>>> A - 1
matrix(
[['0.0', '1.0'],
['2.0', '3.0']])
Of course you can perform matrix multiplication, if the dimensions are
compatible, using ``@`` or ``*``. For clarity, ``@`` is
recommended (`PEP 465 `), because
the meaning of ``*`` is different in many other Python libraries such as NumPy.
>>> A @ B
matrix(
[['8.0', '22.0'],
['14.0', '48.0']])
>>> A * B # same as A @ B
matrix(
[['8.0', '22.0'],
['14.0', '48.0']])
>>> matrix([[1, 2, 3]]) * matrix([[-6], [7], [-2]])
matrix(
[['2.0']])
You can raise powers of square matrices::
>>> A**2
matrix(
[['7.0', '10.0'],
['15.0', '22.0']])
Negative powers will calculate the inverse::
>>> A**-1
matrix(
[['-2.0', '1.0'],
['1.5', '-0.5']])
>>> A * A**-1
matrix(
[['1.0', '1.0842021724855e-19'],
['-2.16840434497101e-19', '1.0']])
Matrix transposition is straightforward::
>>> A = ones(2, 3)
>>> A
matrix(
[['1.0', '1.0', '1.0'],
['1.0', '1.0', '1.0']])
>>> A.T
matrix(
[['1.0', '1.0'],
['1.0', '1.0'],
['1.0', '1.0']])
Norms
.....
Sometimes you need to know how "large" a matrix or vector is. Due to their
multidimensional nature it's not possible to compare them, but there are
several functions to map a matrix or a vector to a positive real number, the
so called norms.
For vectors the p-norm is intended, usually the 1-, the 2- and the oo-norm are
used.
>>> x = matrix([-10, 2, 100])
>>> norm(x, 1)
mpf('112.0')
>>> norm(x, 2)
mpf('100.5186549850325')
>>> norm(x, inf)
mpf('100.0')
Please note that the 2-norm is the most used one, though it is more expensive
to calculate than the 1- or oo-norm.
It is possible to generalize some vector norms to matrix norm::
>>> A = matrix([[1, -1000], [100, 50]])
>>> mnorm(A, 1)
mpf('1050.0')
>>> mnorm(A, inf)
mpf('1001.0')
>>> mnorm(A, 'F')
mpf('1006.2310867787777')
The last norm (the "Frobenius-norm") is an approximation for the 2-norm, which
is hard to calculate and not available. The Frobenius-norm lacks some
mathematical properties you might expect from a norm.
"""
def __init__(self, *args, **kwargs):
self._data = {}
# LU decompostion cache, this is useful when solving the same system
# multiple times, when calculating the inverse and when calculating the
# determinant
self._LU = None
if isinstance(args[0], (list, tuple)):
if not args[0]:
self._rows = 0
self._cols = 0
elif isinstance(args[0][0], (list, tuple)):
# interpret nested list as matrix
A = args[0]
self._rows = len(A)
self._cols = len(A[0])
for i, row in enumerate(A):
for j, a in enumerate(row):
# note: this will call __setitem__ which will call self.ctx.convert() to convert the datatype.
self[i, j] = a
else:
# interpret list as row vector
v = args[0]
self._rows = len(v)
self._cols = 1
for i, e in enumerate(v):
self[i, 0] = e
elif isinstance(args[0], int):
# create empty matrix of given dimensions
if len(args) == 1:
if args[0] < 0:
raise ValueError("expected non-negative int")
self._rows = self._cols = args[0]
else:
if not isinstance(args[1], int):
raise TypeError("expected int")
if args[0] < 0 or args[1] < 0:
raise ValueError("expected non-negative int")
self._rows = args[0]
self._cols = args[1]
elif isinstance(args[0], _matrix):
A = args[0]
self._rows = A._rows
self._cols = A._cols
for i in range(A._rows):
for j in range(A._cols):
self[i, j] = A[i, j]
elif hasattr(args[0], 'tolist'):
A = self.ctx.matrix(args[0].tolist())
self._data = A._data
self._rows = A._rows
self._cols = A._cols
else:
raise TypeError('could not interpret given arguments')
def apply(self, f):
"""
Return a copy of self with the function `f` applied elementwise.
"""
new = self.ctx.matrix(self._rows, self._cols)
for i in range(self._rows):
for j in range(self._cols):
new[i,j] = f(self[i,j])
return new
def __nstr__(self, n=None, **kwargs):
# Build table of string representations of the elements
res = []
# Track per-column max lengths for pretty alignment
maxlen = [0] * self.cols
for i in range(self.rows):
res.append([])
for j in range(self.cols):
if n:
string = self.ctx.nstr(self[i,j], n, **kwargs)
else:
string = str(self[i,j])
res[-1].append(string)
maxlen[j] = max(len(string), maxlen[j])
# Patch strings together
rowsep = '\n'
colsep = ' '
for i, row in enumerate(res):
for j, elem in enumerate(row):
# Pad each element up to maxlen so the columns line up
row[j] = elem.rjust(maxlen[j])
res[i] = "[" + colsep.join(row) + "]"
return rowsep.join(res) if self.rows or self.cols else ''
def __str__(self):
return self.__nstr__()
def _toliststr(self):
"""
Create a list string from a matrix.
If avoid_type: avoid multiple 'mpf's.
"""
# XXX: should be something like self.ctx._types
typ = self.ctx.mpf
s = '['
for i in range(self._rows):
s += '['
for j in range(self._cols):
if not isinstance(self[i,j], typ):
a = repr(self[i,j])
else:
a = "'" + str(self[i,j]) + "'"
s += a + ', '
if s[-1] != '[':
s = s[:-2]
s += '],\n '
if s[-1] != '[':
s = s[:-3]
s += ']'
return s
def tolist(self):
"""
Convert the matrix to a nested list.
"""
return [[self[i,j] for j in range(self._cols)] for i in range(self._rows)]
def __repr__(self):
if self.ctx.pretty:
return self.__str__()
s = 'matrix(\n'
s += self._toliststr() + ')'
return s
def _get_element(self, key):
'''
Fast extraction of the i,j element from the matrix
This function is for private use only because is unsafe:
1. Does not check on the value of key it expects key to be a integer tuple (i,j)
2. Does not check bounds
'''
if key in self._data:
return self._data[key]
else:
return self.ctx.zero
def _set_element(self, key, value):
'''
Fast assignment of the i,j element in the matrix
This function is unsafe:
1. Does not check on the value of key it expects key to be a integer tuple (i,j)
2. Does not check bounds
3. Does not check the value type
4. Does not reset the LU cache
'''
if value: # only store non-zeros
self._data[key] = value
elif key in self._data:
del self._data[key]
def __getitem__(self, key):
'''
Getitem function for mp matrix class with slice index enabled
it allows the following assingments
scalar to a slice of the matrix
B = A[:,2:6]
'''
# Convert vector to matrix indexing
if isinstance(key, int) or isinstance(key,slice):
# only sufficent for vectors
if self._rows == 1:
key = (0, key)
elif self._cols == 1:
key = (key, 0)
else:
raise IndexError('insufficient indices for matrix')
if isinstance(key[0],slice) or isinstance(key[1],slice):
#Rows
if isinstance(key[0],slice):
#Check bounds
if (key[0].start is None or key[0].start >= 0) and \
(key[0].stop is None or key[0].stop <= self._rows+1):
# Generate indices
rows = range(*key[0].indices(self._rows))
else:
raise IndexError('Row index out of bounds')
else:
# Single row
if key[0] >= self._rows:
raise IndexError('Row index out of bounds')
rows = [key[0]]
if(key[0] < 0 and key[0] >= - self._rows):
rows[0] += self._rows
# Columns
if isinstance(key[1],slice):
# Check bounds
if (key[1].start is None or key[1].start >= 0) and \
(key[1].stop is None or key[1].stop <= self._cols+1):
# Generate indices
columns = range(*key[1].indices(self._cols))
else:
raise IndexError('Column index out of bounds')
else:
# Single column
if key[1] >= self._cols:
raise IndexError('Column index out of bounds')
columns = [key[1]]
if(key[1] < 0 and key[1] >= - self._cols):
columns[0] += self._cols
# Create matrix slice
m = self.ctx.matrix(len(rows),len(columns))
# Assign elements to the output matrix
for i,x in enumerate(rows):
for j,y in enumerate(columns):
m._set_element((i,j),self._get_element((x,y)))
return m
else:
row, col = key
if row < 0:
if -row <= self._rows:
row = self._rows + row
else:
raise IndexError('matrix index out of range')
if col < 0:
if -col <= self._cols:
col = self._cols + col
else:
raise IndexError('matrix index out of range')
key = (row, col)
# single element extraction
if key[0] >= self._rows or key[1] >= self._cols:
raise IndexError('matrix index out of range')
if key in self._data:
return self._data[key]
else:
return self.ctx.zero
def __setitem__(self, key, value):
# setitem function for mp matrix class with slice index enabled
# it allows the following assingments
# scalar to a slice of the matrix
# A[:,2:6] = 2.5
# submatrix to matrix (the value matrix should be the same size as the slice size)
# A[3,:] = B where A is n x m and B is n x 1
# Convert vector to matrix indexing
if isinstance(key, int) or isinstance(key,slice):
# only sufficent for vectors
if self._rows == 1:
key = (0, key)
elif self._cols == 1:
key = (key, 0)
else:
raise IndexError('insufficient indices for matrix')
# Slice indexing
if isinstance(key[0],slice) or isinstance(key[1],slice):
# Rows
if isinstance(key[0],slice):
# Check bounds
if (key[0].start is None or key[0].start >= 0) and \
(key[0].stop is None or key[0].stop <= self._rows+1):
# generate row indices
rows = range(*key[0].indices(self._rows))
else:
raise IndexError('Row index out of bounds')
else:
# Single row
rows = [key[0]]
if(key[0] < 0 and key[0] >= - self._rows):
rows[0] += self._rows
# Columns
if isinstance(key[1],slice):
# Check bounds
if (key[1].start is None or key[1].start >= 0) and \
(key[1].stop is None or key[1].stop <= self._cols+1):
# Generate column indices
columns = range(*key[1].indices(self._cols))
else:
raise IndexError('Column index out of bounds')
else:
# Single column
columns = [key[1]]
if(key[1] < 0 and key[1] >= - self._cols):
columns[0] += self._cols
# Assign slice with a scalar
if isinstance(value,self.ctx.matrix):
# Assign elements to matrix if input and output dimensions match
if len(rows) == value.rows and len(columns) == value.cols:
for i,x in enumerate(rows):
for j,y in enumerate(columns):
self._set_element((x,y), value._get_element((i,j)))
else:
raise ValueError('Dimensions do not match')
else:
# Assign slice with scalars
value = self.ctx.convert(value)
for i in rows:
for j in columns:
self._set_element((i,j), value)
else:
# Single element assingment
# Check bounds
row, col = key
if row < 0:
if -row <= self._rows:
row = self._rows + row
else:
raise IndexError('matrix index out of range')
if col < 0:
if -col <= self._cols:
col = self._cols + col
else:
raise IndexError('matrix index out of range')
key = (row, col)
if key[0] >= self._rows or key[1] >= self._cols:
raise IndexError('matrix index out of range')
# Convert and store value
value = self.ctx.convert(value)
if value: # only store non-zeros
self._data[key] = value
elif key in self._data:
del self._data[key]
if self._LU:
self._LU = None
return
def __iter__(self):
for i in range(self._rows):
for j in range(self._cols):
yield self[i,j]
def __mul__(self, other):
if isinstance(other, self.ctx.matrix):
# dot multiplication
if self._cols != other._rows:
raise ValueError('dimensions not compatible for multiplication')
new = self.ctx.matrix(self._rows, other._cols)
for i in range(self._rows):
for j in range(other._cols):
new[i, j] = self.ctx.fdot((self._data[i,k], other._data[k,j])
for k in range(other._rows) if (i,k) in self._data and (k,j) in other._data)
return new
else:
# try scalar multiplication
new = self.ctx.matrix(self._rows, self._cols)
for i in range(self._rows):
for j in range(self._cols):
new[i, j] = other * self[i, j]
return new
def __matmul__(self, other):
return self.__mul__(other)
def __rmul__(self, other):
# assume other is scalar and thus commutative
if isinstance(other, self.ctx.matrix):
raise TypeError("other should not be type of ctx.matrix")
return self.__mul__(other)
def __pow__(self, other):
# avoid cyclic import problems
#from linalg import inverse
if not isinstance(other, int):
raise ValueError('only integer exponents are supported')
if not self._rows == self._cols:
raise ValueError('only powers of square matrices are defined')
n = other
if n == 0:
return self.ctx.eye(self._rows)
if n < 0:
n = -n
neg = True
else:
neg = False
i = n
y = 1
z = self.copy()
while i != 0:
if i % 2 == 1:
y = y * z
z = z*z
i = i // 2
if neg:
y = self.ctx.inverse(y)
return y
def __truediv__(self, other):
# assume other is scalar and do element-wise divison
assert not isinstance(other, self.ctx.matrix)
new = self.ctx.matrix(self._rows, self._cols)
for i in range(self._rows):
for j in range(self._cols):
new[i,j] = self[i,j] / other
return new
def __add__(self, other):
if isinstance(other, self.ctx.matrix):
if not (self._rows == other._rows and self._cols == other._cols):
raise ValueError('incompatible dimensions for addition')
new = self.ctx.matrix(self._rows, self._cols)
for i in range(self._rows):
for j in range(self._cols):
new[i,j] = self[i,j] + other[i,j]
return new
else:
# assume other is scalar and add element-wise
new = self.ctx.matrix(self._rows, self._cols)
for i in range(self._rows):
for j in range(self._cols):
new[i,j] += self[i,j] + other
return new
def __radd__(self, other):
return self.__add__(other)
def __sub__(self, other):
if isinstance(other, self.ctx.matrix) and not (self._rows == other._rows
and self._cols == other._cols):
raise ValueError('incompatible dimensions for subtraction')
return self.__add__(other * (-1))
def __pos__(self):
"""
+M returns a copy of M, rounded to current working precision.
"""
return (+1) * self
def __neg__(self):
return (-1) * self
def __rsub__(self, other):
return -self + other
def __eq__(self, other):
try:
return (self._rows == other._rows and self._cols == other._cols
and self._data == other._data)
except AttributeError:
return NotImplemented
def __len__(self):
if self.rows == 1:
return self.cols
elif self.cols == 1:
return self.rows
else:
return self.rows # do it like numpy
@property
def rows(self):
"""Number of rows."""
return self._rows
@rows.setter
def rows(self, value):
for key in self._data.copy():
if key[0] >= value:
del self._data[key]
self._rows = value
@property
def cols(self):
"""Number of columns."""
return self._cols
@cols.setter
def cols(self, value):
for key in self._data.copy():
if key[1] >= value:
del self._data[key]
self._cols = value
def transpose(self):
new = self.ctx.matrix(self._cols, self._rows)
for i in range(self._rows):
for j in range(self._cols):
new[j,i] = self[i,j]
return new
T = property(transpose)
def conjugate(self):
return self.apply(self.ctx.conj)
def transpose_conj(self):
return self.conjugate().transpose()
H = property(transpose_conj)
def copy(self):
new = self.ctx.matrix(self._rows, self._cols)
new._data = self._data.copy()
return new
__copy__ = copy
def __reduce__(self):
return _make_matrix, (self.tolist(),)
def __array__(self, dtype=None, copy=None):
if copy is not None and not copy:
raise ValueError("`copy=False` isn't supported. A copy is always created.")
from numpy import empty
r = empty((self.rows, self.cols), dtype=dtype)
for i in range(self.rows):
for j in range(self.cols):
r[i, j] = self[i, j]
return r
def column(self, n):
m = self.ctx.matrix(self.rows, 1)
for i in range(self.rows):
m[i] = self[i,n]
return m
class MatrixMethods:
def __init__(ctx):
# XXX: subclass
ctx.matrix = type('matrix', (_matrix,), {})
ctx.matrix.ctx = ctx
ctx.matrix.convert = ctx.convert
def eye(ctx, n, **kwargs):
"""
Create square identity matrix n x n.
"""
A = ctx.matrix(n, **kwargs)
for i in range(n):
A[i,i] = 1
return A
def diag(ctx, diagonal, **kwargs):
"""
Create square diagonal matrix using given list.
Example:
>>> from mpmath import diag, mp
>>> diag([1, 2, 3])
matrix(
[['1.0', '0.0', '0.0'],
['0.0', '2.0', '0.0'],
['0.0', '0.0', '3.0']])
"""
A = ctx.matrix(len(diagonal), **kwargs)
for i in range(len(diagonal)):
A[i,i] = diagonal[i]
return A
def zeros(ctx, *args, **kwargs):
"""
Create matrix m x n filled with zeros.
One given dimension will create square matrix n x n.
Example:
>>> from mpmath import zeros, mp
>>> zeros(2)
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
"""
if len(args) == 1:
m = n = args[0]
elif len(args) == 2:
m = args[0]
n = args[1]
else:
raise TypeError('zeros expected at most 2 arguments, got %i' % len(args))
A = ctx.matrix(m, n, **kwargs)
for i in range(m):
for j in range(n):
A[i,j] = 0
return A
def ones(ctx, *args, **kwargs):
"""
Create matrix m x n filled with ones.
One given dimension will create square matrix n x n.
Example:
>>> from mpmath import ones, mp
>>> ones(2)
matrix(
[['1.0', '1.0'],
['1.0', '1.0']])
"""
if len(args) == 1:
m = n = args[0]
elif len(args) == 2:
m = args[0]
n = args[1]
else:
raise TypeError('ones expected at most 2 arguments, got %i' % len(args))
A = ctx.matrix(m, n, **kwargs)
for i in range(m):
for j in range(n):
A[i,j] = 1
return A
def hilbert(ctx, m, n=None):
"""
Create (pseudo) hilbert matrix m x n.
One given dimension will create hilbert matrix n x n.
The matrix is very ill-conditioned and symmetric, positive definite if
square.
"""
if n is None:
n = m
A = ctx.matrix(m, n)
for i in range(m):
for j in range(n):
A[i,j] = ctx.one / (i + j + 1)
return A
def randmatrix(ctx, m, n=None, min=0, max=1, **kwargs):
"""
Create a random m x n matrix.
All values are >= min and >> from mpmath import randmatrix
>>> randmatrix(2) # doctest:+SKIP
matrix(
[['0.53491598236191806', '0.57195669543302752'],
['0.85589992269513615', '0.82444367501382143']])
"""
if not n:
n = m
A = ctx.matrix(m, n, **kwargs)
for i in range(m):
for j in range(n):
A[i,j] = ctx.rand() * (max - min) + min
return A
def swap_row(ctx, A, i, j):
"""
Swap row i with row j.
"""
if i == j:
return
if isinstance(A, ctx.matrix):
for k in range(A.cols):
A[i,k], A[j,k] = A[j,k], A[i,k]
elif isinstance(A, list):
A[i], A[j] = A[j], A[i]
else:
raise TypeError('could not interpret type')
def extend(ctx, A, b):
"""
Extend matrix A with column b and return result.
"""
if not isinstance(A, ctx.matrix):
raise TypeError("A should be a type of ctx.matrix")
if A.rows != len(b):
raise ValueError("Value should be equal to len(b)")
A = A.copy()
A.cols += 1
for i in range(A.rows):
A[i, A.cols-1] = b[i]
return A
def norm(ctx, x, p=2):
r"""
Gives the entrywise `p`-norm of an iterable *x*, i.e. the vector norm
`\left(\sum_k |x_k|^p\right)^{1/p}`, for any given `1 \le p \le \infty`.
Special cases:
If *x* is not iterable, this just returns ``absmax(x)``.
``p=1`` gives the sum of absolute values.
``p=2`` is the standard Euclidean vector norm.
``p=inf`` gives the magnitude of the largest element.
For *x* a matrix, ``p=2`` is the Frobenius norm.
For operator matrix norms, use :func:`~mpmath.mnorm` instead.
You can use the string 'inf' as well as float('inf') or mpf('inf')
to specify the infinity norm.
**Examples**
>>> from mpmath import matrix, norm, inf
>>> x = matrix([-10, 2, 100])
>>> norm(x, 1)
mpf('112.0')
>>> norm(x, 2)
mpf('100.5186549850325')
>>> norm(x, inf)
mpf('100.0')
"""
try:
iter(x)
except TypeError:
return ctx.absmax(x)
if type(p) is not int:
p = ctx.convert(p)
if p == ctx.inf:
return max(ctx.absmax(i) for i in x)
elif p == 1:
return ctx.fsum(x, absolute=1)
elif p == 2:
return ctx.sqrt(ctx.fsum(x, absolute=1, squared=1))
elif p > 1:
return ctx.nthroot(ctx.fsum(abs(i)**p for i in x), p)
else:
raise ValueError('p has to be >= 1')
def mnorm(ctx, A, p=1):
r"""
Gives the matrix (operator) `p`-norm of A. Currently ``p=1`` and ``p=inf``
are supported:
``p=1`` gives the 1-norm (maximal column sum)
``p=inf`` gives the `\infty`-norm (maximal row sum).
You can use the string 'inf' as well as float('inf') or mpf('inf')
``p=2`` (not implemented) for a square matrix is the usual spectral
matrix norm, i.e. the largest singular value.
``p='f'`` (or 'F', 'fro', 'Frobenius, 'frobenius') gives the
Frobenius norm, which is the elementwise 2-norm. The Frobenius norm is an
approximation of the spectral norm and satisfies
.. math ::
\frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F
The Frobenius norm lacks some mathematical properties that might
be expected of a norm.
For general elementwise `p`-norms, use :func:`~mpmath.norm` instead.
**Examples**
>>> from mpmath import matrix, mnorm, inf
>>> A = matrix([[1, -1000], [100, 50]])
>>> mnorm(A, 1)
mpf('1050.0')
>>> mnorm(A, inf)
mpf('1001.0')
>>> mnorm(A, 'F')
mpf('1006.2310867787777')
"""
A = ctx.matrix(A)
if type(p) is not int:
if type(p) is str and 'frobenius'.startswith(p.lower()):
return ctx.norm(A, 2)
p = ctx.convert(p)
m, n = A.rows, A.cols
if p == 1:
return max((ctx.fsum((A[i,j] for i in range(m)), absolute=1) for j in range(n)), default=0)
elif p == ctx.inf:
return max((ctx.fsum((A[i,j] for j in range(n)), absolute=1) for i in range(m)), default=0)
else:
raise NotImplementedError("matrix p-norm for arbitrary p")