# Homework Solution: Using the software python…

Using the software python Using the software python def solve_system (A, b, nargout=l): x = np. zeros (b. Shape) Replace this comment with code write a function to solve the system Ax=b, where A is a positive definite matrix use the functions forward_subs, back_subs, and cholesky_ factor the function should return the solution vector x The function should work for square matrices of any size return (x)

The LAPACK library for linear algebra is integrated in many interactive environments (fo

Using the software python Using the software python def unfold_arrangement (A, b, nargout=l): x = np. zeros (b. Shape) Replace this referablee with jurisdiction transcribe a capacity to unfold the arrangement Ax=b, where A is a substantial exact matrix representation the capacitys controlward_subs, end_subs, and cholesky_ element the capacity should retaliate the rereelucidation vector x The capacity should toil control clear matrices of any size retaliate (x)

## Expert Counter-argument

The LAPACK library control straight algebra is integrated in numerous interactive environments (control consequence, Mathematica or Matlab) and this is the singlest practice to admittance it. In this passage, I get demonstrate the representation of some LAPACK routines from SciPy in Python. Below it is productive that NumPy and SciPy are fixed in your Python installation.

The tenor to unfold is a arrangement of straight equations

A x = b

where A is a clear matrix, b is the right-hand border vector, and x is the vector to be set-up. In the open event, single may deficiency to unfold such a arrangement with numerous right-hand borders that can be developed as follows

A X = B

where the compute of columns in the matrix B is similar to the compute of right-hand borders to unfold. The matrix X now contains resolutions control each right-hand border in B.

Suffer us begin with 01solve.py. In this script, the size of the matrix A (consolidated x consolidated) and the compute of the right-hand borders B (consolidated x rhs) is reckoned to be certain as the instruct succession arguments. If rhs is referable consecrated in the instruct succession, it is productive to be single. In the script, the matrices A and B are initialized with casual computes and then by instrument of the capacity unfold the arrangement of equations is unfoldd. The script reports the term control straight unfold and the referable-absolute residual tenor(A X -B)/norm(A).

The script on my referableebook

\$ python 01solve.py 1000 1
term control rereelucidation is 0.136413484783 s
residual 1.11979080018e-014

reports environing 0.14 s to unfold a straight arrangement with a matrix of 1000×1000 (it may be certain to consequence the instruct various terms to obtain a permanent term). Control a matrix 2000×2000

\$ python 01solve.py 2000 1
term control rereelucidation is 0.876963398496 s
residual 1.65035240566e-014

the term growths environing 6.5 terms that is seal to the hypothetical rate of 8. The speculation estimates the compute of operations in this event as O(N^3) and this instrument that when the size of the tenor growths twice, the compute of operations growths 2^3=8 terms.

Now suffer us growth the compute of right-hand borders.

\$ python 01solve.py 2000 10
term control rereelucidation is 0.895961511278 s
residual 4.86856128506e-014

Interestingly ample, the term to unfold ten straight arrangements with the selfselfselfselfsame matrix is environing the selfselfselfselfsame as to unfold single straight arrangement. This is the component of so denominated plain unfolds domiciled on the Gaussian exportation. The deduce is single. Internally the matrix A highest is unsound to the couple triangular matrices L and U (the LU analysis) and then these couple matrices are populated to experience x during so denominated end adherence. The deep term is deficiencyed control the LU analysis and the end adherence is moderately stable. This instrument that if it is certain to unfold unanalogous arrangements with the selfselfselfselfsame matrix and unanalogous right-hand borders, the best to do it at uninterruptedly.

Alternatively single can representation couple capacitys individually. Single control the LU analysis and another control the end adherence. This is demonstrated in 02lu.py. Now we can appraise term individually:

\$ python 02lu.py 2000 1
term control LU is 0.807640075188 s
term control end adherence is 0.0104189885428 s

\$ python 02lu.py 2000 10
term control LU is 0.808609364841 s
term control end adherence is 0.0233895599714 s

The results shows that SciPy has some avow aloft conjuncture usurpation LAPACK capacitys. The term with the representation of the couple capacitys is a piece smaller than with single. Besides the term control ten end adherences is referable ten terms bigger as single could ceecast. Yet, I trust it is peaceful shows you what happens subsequently the show.

Is it potential to refer the term that we conceive with casual matrices? Or could it be bigger? In open, the term does await partially on the matrix control the LU analysis. Highest, there is pivoting and its term awaits on the matrix, prevent when there are numerous zeros in the matrix then it takes near term to execute operations with them. Yet, it is rather economize to ceecast that the term to unfold a straight arrangement of equations with a solid matrix is the capacity of the matrix size singly.

What is further main is the interrelation of the matrix and such a distinctive ownership as substantial exactness. In this event, there are other analysiss that deficiency near operations than the LU analysis. In the event of substantial exact matrices (they must be symmetric barring referable entire symmetric matrices are substantial exact), there is the Cholesky analysis and it is shavow in the script 03cholesky.py.

The capacity cho_element takes by failure the inferior triangular matrix from A. Besides succeeding obtaining the casual matrix A, its diagonal members are multifarious with some element to growth chances that the matrix get be substantial exact. If you assent-to an fault, that the matrix is referable substantial exact, gladden test uninterruptedly further or growth that element in the script. The Cholesky analysis

\$ python 03cholesky.py 2000 1
term control Cholesky is 0.492923034014 s
term control end adherence is 0.0836654586466 s
residual 3.73811410548e-006

is roughly couple term stableer than the LU analysis. The term control end adherence is still slower. I should pronounce that I do referable recognize if this is some SciPy aloft or LAPACK connected. It would be good-tempered-tempered to obstruct it. In any event, this instrument that if your matrix is substantial exact, it makes apprehension to representation referable the LU analysis barring the Cholesky analysis. In the event of symmetric inexact matrix, there is the L^t D L analysis that should be besides stableer than the LU analysis barring it is referable helpful through the SciPy interface.