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How to show Schulz's method converges $Q$-quadratically?

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1

$begingroup$

Let $A in mathbb{R}^{n times n}$ be an invertible matrix. Consider the Schulz's method for fast computation of the inverse of $A$ given by:

$$
X_{k+1} = X_k + X_k (I -AX_k)
$$

where each $X_k in mathbb{R}^{n times n}$, and $X_0$ is such that $|A- AX_0| <1$.

Define $E_k=I -AX_k$. Then we know $E_{k+1}=E_k^2$ for all $k$, show that the sequence $(E_k)$ converges to zero matrix $Q$-quadratically.

linear-algebra numerical-linear-algebra

share|cite|improve this question

edited yesterday

Jean Marie

30.5k42154

asked 2 days ago

SepideSepide

4188

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  math.stackexchange.com/questions/1058476/…
  $endgroup$
  – Jean Marie
  yesterday
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $A in mathbb{R}^{n times n}$ be an invertible matrix. Consider the Schulz's method for fast computation of the inverse of $A$ given by:

$$
X_{k+1} = X_k + X_k (I -AX_k)
$$

where each $X_k in mathbb{R}^{n times n}$, and $X_0$ is such that $|A- AX_0| <1$.

Define $E_k=I -AX_k$. Then we know $E_{k+1}=E_k^2$ for all $k$, show that the sequence $(E_k)$ converges to zero matrix $Q$-quadratically.

linear-algebra numerical-linear-algebra

share|cite|improve this question

edited yesterday

Jean Marie

30.5k42154

asked 2 days ago

SepideSepide

4188

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  math.stackexchange.com/questions/1058476/…
  $endgroup$
  – Jean Marie
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

Let $A in mathbb{R}^{n times n}$ be an invertible matrix. Consider the Schulz's method for fast computation of the inverse of $A$ given by:

$$
X_{k+1} = X_k + X_k (I -AX_k)
$$

where each $X_k in mathbb{R}^{n times n}$, and $X_0$ is such that $|A- AX_0| <1$.

Define $E_k=I -AX_k$. Then we know $E_{k+1}=E_k^2$ for all $k$, show that the sequence $(E_k)$ converges to zero matrix $Q$-quadratically.

linear-algebra numerical-linear-algebra

share|cite|improve this question

edited yesterday

Jean Marie

30.5k42154

asked 2 days ago

SepideSepide

4188

$endgroup$

share|cite|improve this question

edited yesterday

Jean Marie

30.5k42154

asked 2 days ago

SepideSepide

4188

share|cite|improve this question

edited yesterday

Jean Marie

30.5k42154

asked 2 days ago

SepideSepide

4188

edited yesterday

Jean Marie

30.5k42154

edited yesterday

Jean Marie

30.5k42154

asked 2 days ago

SepideSepide

4188

asked 2 days ago

SepideSepide

4188