B = Hermitian positive-definite matrix A into matrix V, so that V* V* = A, where V* denotes the conjugate transpose of L. The dimensions of the matrix A must match. This method is mainly used for numeric solution of linear equations Ax = b. example: Input matrix A: [[ 4, 12, -16], [ 12, 37, -43], [-16, -43, 98]] Result:
B = Hermitian positive-definite matrix A into matrix V, so that V* V* = A, where V* denotes the conjugate transpose of L. The dimensions of the matrix A must match. This method is mainly used for numeric solution of linear equations Ax = b. example: Input matrix A: [[ 4, 12, -16], [ 12, 37, -43], [-16, -43, 98]] Result:
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
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Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Cholesky matrix decomposition is used to find the decomposition of a
Hermitian positive-definite matrix A
into matrix V, so that V* V* = A, where V* denotes the conjugate
transpose of L.
The dimensions of the matrix A must match.
This method is mainly used for numeric solution of linear equations Ax = b.
example:
Input matrix A:
[[ 4, 12, -16],
[ 12, 37, -43],
[-16, -43, 98]]
Result:
[[2.0, 0.0, 0.0],
[6.0, 1.0, 0.0],
[-8.0, 5.0, 3.0]]
Time complexity of this algorithm is O(n^3), specifically about (n^3)/3
*1*1#
import math
def cholesky_decomposition (A):
******
:param A: Hermitian positive-definite matrix of type List[List[float]]
:return: matrix of type List[List[float]] if A can be decomposed,
otherwise None
n = len(A)
for ai in A:
if len(ai) != n:
return None
V [[0.0] *n for in range(n)]
for j in range(n):
sum_diagonal_element = 0
for k in range(j):
sum_diagonal_element = sum_diagonal_element + math.pow(V[j][k], 2)
sum_diagonal_element = A[j][j] sum_diagonal_element
if sum_diagonal_element <= 0:
return None
V[j][j] = math.pow(sum_diagonal_element, 0.5)
for i in range(j+1, n):
sum_other_element = 0.
for k in range(j):
sum_other element += V[i] [k]*V[j][k]
V[i][j] =(A[i][j] sum_other_element)/V[j][j]
return V](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e38f750-a3b6-41fa-9e5a-77c8be67512c%2F8e652394-5044-4c42-87d7-e77dafb894ad%2Fgeakg6_processed.jpeg&w=3840&q=75)
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Cholesky matrix decomposition is used to find the decomposition of a
Hermitian positive-definite matrix A
into matrix V, so that V* V* = A, where V* denotes the conjugate
transpose of L.
The dimensions of the matrix A must match.
This method is mainly used for numeric solution of linear equations Ax = b.
example:
Input matrix A:
[[ 4, 12, -16],
[ 12, 37, -43],
[-16, -43, 98]]
Result:
[[2.0, 0.0, 0.0],
[6.0, 1.0, 0.0],
[-8.0, 5.0, 3.0]]
Time complexity of this algorithm is O(n^3), specifically about (n^3)/3
*1*1#
import math
def cholesky_decomposition (A):
******
:param A: Hermitian positive-definite matrix of type List[List[float]]
:return: matrix of type List[List[float]] if A can be decomposed,
otherwise None
n = len(A)
for ai in A:
if len(ai) != n:
return None
V [[0.0] *n for in range(n)]
for j in range(n):
sum_diagonal_element = 0
for k in range(j):
sum_diagonal_element = sum_diagonal_element + math.pow(V[j][k], 2)
sum_diagonal_element = A[j][j] sum_diagonal_element
if sum_diagonal_element <= 0:
return None
V[j][j] = math.pow(sum_diagonal_element, 0.5)
for i in range(j+1, n):
sum_other_element = 0.
for k in range(j):
sum_other element += V[i] [k]*V[j][k]
V[i][j] =(A[i][j] sum_other_element)/V[j][j]
return V
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