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#######################################################
# Copyright (c) 2015, ArrayFire
# All rights reserved.
#
# This file is distributed under 3-clause BSD license.
# The complete license agreement can be obtained at:
# http://arrayfire.com/licenses/BSD-3-Clause
########################################################
"""
Dense Linear Algebra functions (solve, inverse, etc).
"""
from
.
library
import
*
from
.
array
import
*
def
lu
(
A
):
"""
LU decomposition.
Parameters
----------
A: af.Array
A 2 dimensional arrayfire array.
Returns
-------
(L,U,P): tuple of af.Arrays
- L - Lower triangular matrix.
- U - Upper triangular matrix.
- P - Permutation array.
Note
----
The original matrix `A` can be reconstructed using the outputs in the following manner.
>>> A[P, :] = af.matmul(L, U)
"""
L
=
Array
()
U
=
Array
()
P
=
Array
()
safe_call
(
backend
.
get
().
af_lu
(
c_pointer
(
L
.
arr
),
c_pointer
(
U
.
arr
),
c_pointer
(
P
.
arr
),
A
.
arr
))
return
L
,
U
,
P
def
lu_inplace
(
A
,
pivot
=
"lapack"
):
"""
In place LU decomposition.
Parameters
----------
A: af.Array
- a 2 dimensional arrayfire array on entry.
- Contains L in the lower triangle on exit.
- Contains U in the upper triangle on exit.
Returns
-------
P: af.Array
- Permutation array.
Note
----
This function is primarily used with `af.solve_lu` to reduce computations.
"""
P
=
Array
()
is_pivot_lapack
=
False
if
(
pivot
==
"full"
)
else
True
safe_call
(
backend
.
get
().
af_lu_inplace
(
c_pointer
(
P
.
arr
),
A
.
arr
,
is_pivot_lapack
))
return
P
def
qr
(
A
):
"""
QR decomposition.
Parameters
----------
A: af.Array
A 2 dimensional arrayfire array.
Returns
-------
(Q,R,T): tuple of af.Arrays
- Q - Orthogonal matrix.
- R - Upper triangular matrix.
- T - Vector containing additional information to solve a least squares problem.
Note
----
The outputs of this funciton have the following properties.
>>> A = af.matmul(Q, R)
>>> I = af.matmulNT(Q, Q) # Identity matrix
"""
Q
=
Array
()
R
=
Array
()
T
=
Array
()
safe_call
(
backend
.
get
().
af_qr
(
c_pointer
(
Q
.
arr
),
c_pointer
(
R
.
arr
),
c_pointer
(
T
.
arr
),
A
.
arr
))
return
Q
,
R
,
T
def
qr_inplace
(
A
):
"""
In place QR decomposition.
Parameters
----------
A: af.Array
- a 2 dimensional arrayfire array on entry.
- Packed Q and R matrices on exit.
Returns
-------
T: af.Array
- Vector containing additional information to solve a least squares problem.
Note
----
This function is used to save space only when `R` is required.
"""
T
=
Array
()
safe_call
(
backend
.
get
().
af_qr_inplace
(
c_pointer
(
T
.
arr
),
A
.
arr
))
return
T
def
cholesky
(
A
,
is_upper
=
True
):
"""
Cholesky decomposition
Parameters
----------
A: af.Array
A 2 dimensional, symmetric, positive definite matrix.
is_upper: optional: bool. default: True
Specifies if output `R` is upper triangular (if True) or lower triangular (if False).
Returns
-------
(R,info): tuple of af.Array, int.
- R - triangular matrix.
- info - 0 if decomposition sucessful.
Note
----
The original matrix `A` can be reconstructed using the outputs in the following manner.
>>> A = af.matmulNT(R, R) #if R is upper triangular
"""
R
=
Array
()
info
=
c_int_t
(
0
)
safe_call
(
backend
.
get
().
af_cholesky
(
c_pointer
(
R
.
arr
),
c_pointer
(
info
),
A
.
arr
,
is_upper
))
return
R
,
info
.
value
def
cholesky_inplace
(
A
,
is_upper
=
True
):
"""
In place Cholesky decomposition.
Parameters
----------
A: af.Array
- a 2 dimensional, symmetric, positive definite matrix.
- Trinangular matrix on exit.
is_upper: optional: bool. default: True.
Specifies if output `R` is upper triangular (if True) or lower triangular (if False).
Returns
-------
info : int.
0 if decomposition sucessful.
"""
info
=
c_int_t
(
0
)
safe_call
(
backend
.
get
().
af_cholesky_inplace
(
c_pointer
(
info
),
A
.
arr
,
is_upper
))
return
info
.
value
def
solve
(
A
,
B
,
options
=
MATPROP
.
NONE
):
"""
Solve a system of linear equations.
Parameters
----------
A: af.Array
A 2 dimensional arrayfire array representing the coefficients of the system.
B: af.Array
A 1 or 2 dimensional arrayfire array representing the constants of the system.
options: optional: af.MATPROP. default: af.MATPROP.NONE.
- Additional options to speed up computations.
- Currently needs to be one of `af.MATPROP.NONE`, `af.MATPROP.LOWER`, `af.MATPROP.UPPER`.
Returns
-------
X: af.Array
A 1 or 2 dimensional arrayfire array representing the unknowns in the system.
"""
X
=
Array
()
safe_call
(
backend
.
get
().
af_solve
(
c_pointer
(
X
.
arr
),
A
.
arr
,
B
.
arr
,
options
.
value
))
return
X
def
solve_lu
(
A
,
P
,
B
,
options
=
MATPROP
.
NONE
):
"""
Solve a system of linear equations, using LU decomposition.
Parameters
----------
A: af.Array
- A 2 dimensional arrayfire array representing the coefficients of the system.
- This matrix should be decomposed previously using `lu_inplace(A)`.
P: af.Array
- Permutation array.
- This array is the output of an earlier call to `lu_inplace(A)`
B: af.Array
A 1 or 2 dimensional arrayfire array representing the constants of the system.
Returns
-------
X: af.Array
A 1 or 2 dimensional arrayfire array representing the unknowns in the system.
"""
X
=
Array
()
safe_call
(
backend
.
get
().
af_solve_lu
(
c_pointer
(
X
.
arr
),
A
.
arr
,
P
.
arr
,
B
.
arr
,
options
.
value
))
return
X
def
inverse
(
A
,
options
=
MATPROP
.
NONE
):
"""
Invert a matrix.
Parameters
----------
A: af.Array
- A 2 dimensional arrayfire array
options: optional: af.MATPROP. default: af.MATPROP.NONE.
- Additional options to speed up computations.
- Currently needs to be one of `af.MATPROP.NONE`.
Returns
-------
AI: af.Array
- A 2 dimensional array that is the inverse of `A`
Note
----
`A` needs to be a square matrix.
"""
AI
=
Array
()
safe_call
(
backend
.
get
().
af_inverse
(
c_pointer
(
AI
.
arr
),
A
.
arr
,
options
.
value
))
return
AI
def
pinverse
(
A
,
tol
=
1E-6
,
options
=
MATPROP
.
NONE
):
"""
Find pseudo-inverse(Moore-Penrose) of a matrix.
Parameters
----------
A: af.Array
- A 2 dimensional arrayfire input matrix array
tol: optional: scalar. default: 1E-6.
- Tolerance for calculating rank
options: optional: af.MATPROP. default: af.MATPROP.NONE.
- Currently needs to be `af.MATPROP.NONE`.
- Additional options may speed up computation in the future
Returns
-------
AI: af.Array
- A 2 dimensional array that is the pseudo-inverse of `A`
Note
----
This function is not supported in GFOR
"""
AI
=
Array
()
safe_call
(
backend
.
get
().
af_pinverse
(
c_pointer
(
AI
.
arr
),
A
.
arr
,
c_double_t
(
tol
),
options
.
value
))
return
AI
def
rank
(
A
,
tol
=
1E-5
):
"""
Rank of a matrix.
Parameters
----------
A: af.Array
- A 2 dimensional arrayfire array
tol: optional: scalar. default: 1E-5.
- Tolerance for calculating rank
Returns
-------
r: int
- Rank of `A` within the given tolerance
"""
r
=
c_uint_t
(
0
)
safe_call
(
backend
.
get
().
af_rank
(
c_pointer
(
r
),
A
.
arr
,
c_double_t
(
tol
)))
return
r
.
value
def
det
(
A
):
"""
Determinant of a matrix.
Parameters
----------
A: af.Array
- A 2 dimensional arrayfire array
Returns
-------
res: scalar
- Determinant of the matrix.
"""
re
=
c_double_t
(
0
)
im
=
c_double_t
(
0
)
safe_call
(
backend
.
get
().
af_det
(
c_pointer
(
re
),
c_pointer
(
im
),
A
.
arr
))
re
=
re
.
value
im
=
im
.
value
return
re
if
(
im
==
0
)
else
re
+
im
*
1j
def
norm
(
A
,
norm_type
=
NORM
.
EUCLID
,
p
=
1.0
,
q
=
1.0
):
"""
Norm of an array or a matrix.
Parameters
----------
A: af.Array
- A 1 or 2 dimensional arrayfire array
norm_type: optional: af.NORM. default: af.NORM.EUCLID.
- Type of norm to be calculated.
p: scalar. default 1.0.
- Used only if `norm_type` is one of `af.NORM.VECTOR_P`, `af.NORM_MATRIX_L_PQ`
q: scalar. default 1.0.
- Used only if `norm_type` is `af.NORM_MATRIX_L_PQ`
Returns
-------
res: scalar
- norm of the input
"""
res
=
c_double_t
(
0
)
safe_call
(
backend
.
get
().
af_norm
(
c_pointer
(
res
),
A
.
arr
,
norm_type
.
value
,
c_double_t
(
p
),
c_double_t
(
q
)))
return
res
.
value
def
svd
(
A
):
"""
Singular Value Decomposition
Parameters
----------
A: af.Array
A 2 dimensional arrayfire array.
Returns
-------
(U,S,Vt): tuple of af.Arrays
- U - A unitary matrix
- S - An array containing the elements of diagonal matrix
- Vt - A unitary matrix
Note
----
- The original matrix `A` is preserved and additional storage space is required for decomposition.
- If the original matrix `A` need not be preserved, use `svd_inplace` instead.
- The original matrix `A` can be reconstructed using the outputs in the following manner.
>>> Smat = af.diag(S, 0, False)
>>> A_recon = af.matmul(af.matmul(U, Smat), Vt)
"""
U
=
Array
()
S
=
Array
()
Vt
=
Array
()
safe_call
(
backend
.
get
().
af_svd
(
c_pointer
(
U
.
arr
),
c_pointer
(
S
.
arr
),
c_pointer
(
Vt
.
arr
),
A
.
arr
))
return
U
,
S
,
Vt
def
svd_inplace
(
A
):
"""
Singular Value Decomposition
Parameters
----------
A: af.Array
A 2 dimensional arrayfire array.
Returns
-------
(U,S,Vt): tuple of af.Arrays
- U - A unitary matrix
- S - An array containing the elements of diagonal matrix
- Vt - A unitary matrix
Note
----
- The original matrix `A` is not preserved.
- If the original matrix `A` needs to be preserved, use `svd` instead.
- The original matrix `A` can be reconstructed using the outputs in the following manner.
>>> Smat = af.diag(S, 0, False)
>>> A_recon = af.matmul(af.matmul(U, Smat), Vt)
"""
U
=
Array
()
S
=
Array
()
Vt
=
Array
()
safe_call
(
backend
.
get
().
af_svd_inplace
(
c_pointer
(
U
.
arr
),
c_pointer
(
S
.
arr
),
c_pointer
(
Vt
.
arr
),
A
.
arr
))
return
U
,
S
,
Vt
def
is_lapack_available
():
"""
Function to check if the arrayfire library was built with lapack support.
"""
res
=
c_bool_t
(
False
)
safe_call
(
backend
.
get
().
af_is_lapack_available
(
c_pointer
(
res
)))
return
res
.
value
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