ericdarve / numerical_linear_algebra Goto Github PK

In the presentation of the SVD on page 66, the theorem writes A = U\Sigma V^H, but below the page the notation is switched to V^T instead.

$$x + y = x + y$$ should be $$x + y = y + x$$

--
Honglin Yuan

"The scalars in this book will be real or complex numbers, by (should be but?) generally it is enough if they belong to a field."

Best,
Fabian Boemer

The denominator in the third equation should be (b-a)
(Since we are considering lambda =lamda(H) = H in the scalar case)

—
Honglin Yuan

In Line 6,

"since we have q_1 = r_{11}a_1"
should be
"since we have a_1 = r_{11}q_1"

--
Honglin Yuan

The block singular vector is defined as $q_i$, but when demonstrating how it interacts with the block matrix $B$, the explanation refers to "$x_i$":

"Then $x_i^TBx_i = \pm\sigma_i$."

It should read:

"Then $q_i^TBq_i = \pm\sigma_i$"

for consistency of notation.

--Martin Lindsey

On page 209, at the top when discussing diagonally dominant, does "i" in the statement "for i = 1,2,...n" need to be changed to "j" since we are summing the entries along the ith row from j = 1,...n ?

The error in the estimated eigenvalue decays at (lambda_2/lambda_1)^2k instead of k?

The middle diagram shows how Q1 acts on the first column of A. The norm of a1 should be a 2-norm instead of 1-norm.

a[2,:] should return the second row vector of a.
It should be [2.0 2.71828], instead of [2.0, 2.71828] (which is a column vector).

--
Honglin Yuan

"After that, the convergence will look like these eigenvalues no longer exist such that the condition number of A is apparently reduced, resulting in faster convergence."

Does this mean that once the well-separated eigenvalues have good estimates, they will no longer contribute (i.e., it will seem like A has a lower condition number), which will then accelerate convergence?
If so, I think the current phrasing in the course reader seems confusing.

"However, notice that if we permute rows 3 and 4"

Best,
Fabian Boemer

Second-to-last paragraph:
"Sparse matrices are matrices where most of the entries are equal to zero."

Best,
Fabian Boemer

The second proof has incorrect equations (far as I can tell). They should read:

Should be "Cayley-Hamilton theorem"

Convergence rate of eigenvalues is twice as fast for symmetric A than for non-symmetric A. The code generates a symmetric matrix.

"We prove the scalar case first get the general idea of the proof" should probably read "We prove the scalar case first to get the general idea of the proof".

Glitch on pdf rendering, red text overlapping the matrix.

On the first line of minimizer polynomial, it states "The zeros of pk are our eigenvalue approximations.".

Should it be zero root? or roots? it seems to be confusing what the sentence actually means

In Exercise 2.2, shouldn't it be "use A(t) = tB", instead of "use A(t) = exp(tB)", since in the equation we have det(exp A)?
But then proving starting from det(A) to this equation will not be possible.

In the proof of necessary condition of convergence of SOR:
The second inequality should be
|lambda_max(G)| >= |1-w|

The equality may hold.

—
Honglin Yuan

"The following geometric argument shows that v = x - ||x||e1 will do the trick." instead of
"The following geometric argument shows that v = x + ||x||e1 will do the trick."

Hi,

  At page 56, the first sentence of first block "interpretation of physical systems to understand the important of eignevalues".

  I think it should be "importance"  instead of "important".  This may be a minor issue, but may bother some readers.

  I like you book!

Lei Fang

On page 207, in theorem 8.1, the Spectral radius is defined as ”the largest eigenvalue of the matrix”

Should be the maximum modulus of the eigenvalues? (Since this evals may not be real)

— Honglin Yuan

The second box says null space of A, it should be of A_T

At the last line, I guess it should be $O(r n^2)$

On page 93, in the 4th row from bottom,
u = 1/2 (distance between 1 and the next largest floating point)
"largest" does not make sense.

— Honglin Yuan

Equation (4.1), for entries such as (g_1)_21 etc. I think the subscript outside the bracket should simply be 2 (a single number) since g_i is a column vector?

The "@code failed to converge" in the first code block is indented too far.

Best,
Fabian Boemer

On pg 91 you have the equation

1.5703125 = 1/2 + 1/16 + 1/128

I think you forgot to add a 1. The right hand side only gives the value after the decimal.

In the caption of Figure 3.4, "The axes are scales (should be scaled?) by A", and
"The unit circle is shown with a dashed line".

Best,
Fabian Boemer

Should the first sigma_max be sigma_min?

The first line in the proof at the bottom of the page is missing an I after the first mu-bar, in the substitution for U_2 R_2.

In the third line of page 154, "But, we know that the eigenvalues of (A−μI)^(−1) are the inverses of the eigenvalues of A−μI." The current wording (with the bold) could cause confusion, since A−μI is a matrix.

Best,
Fabian Boemer

On page 209, at the top when discussing diagonally dominant, does "i" in the statement "for i = 1,2,...n" need to be changed to "j" since we are summing the entries along the ith row from j = 1,...n ?

At the bottom, when defining the matrix $G_1$, the text switches to a lowercase $g_1$ when defining entries:

You can check that

1.5703125 =...

should be 0.5703125=...

Hi,

At page 69, the second equation. A^(2k+1)v_1=Au_1.

Thank you!

Lei Fang (from cme302 class)

The red boxes end up on the bad rows/columns. Goal was to emphasize last column of L and last row of U.

"The notebook QR_Givens.py shows..."
This seems like the only time there is a ".py" (as well as Exercise 2.1 on the same page) instead of a ".ipynb"

Not a typo per-se, but seems inconsistent with the rest of the book.

Best,
Fabian Boemer

"1. Pick M such that linear systems Mz = d are be easy to solve" should read "...are easy to solve".

In block "Normal equation method"

A^b should be "A^Tb

--
Honglin Yuan

(This is a minor issue, but I think it will be clearer if you can rephrase it. I discuss it with Eric during today's OH):

"The numerical approximations of L and U, $$\tilde{L}$$ and $$\tilde{U}$$, are in fact very accurate."

Word missing somewhere here:

"The main observation is that we are going the matrix T"...

"Then Q1.T.a1 should be parallel to e1. Since Q1.T does not change the norm of a1.." instead of "Then "Q1.a1 should be parallel to e1. Since Q1 does not change the norm of a1.."

It seems that either f(x) and f(\tilde{x}) or x and (\tilde{x}) should flip.

--
Honglin Yuan

In the second code block, "@ Exceptional shift " is indented too far.

Best,
Fabian Boemer

"Σ has zeros in rows r + 1 to m" (page 148)
Not sure what is proper gramatically, but it should be consistent throughout the text.
For comparison,

• "R ̃ is m × n and has zeros in rows n + 1 to m." (page 147)
• "the rows r + 1 to n of U will" (page 109, Exercise 4.2)

(All of the following potential improvements are based on the discussion with Eric during today's office hour. )

1. P152:
   To make it clear, it's better to add more explanations on how we come from equation (6.1) to the next equation:A^k \approx \lambda_1^k x_1 y_1^T.
2. P153:
   After equation
   A^{k} q^{(0)} = \lambda^k_1 \alpha_1 x_1 +\cdots + \lambda^k_n \alpha_n x_n,
   it's better to add one more line:
   = \lambda^k_1 (\alpha_1 x_1 +(\frac{ \alpha_2}{ \alpha_1})^k x_2 \cdots + (\frac{ \alpha_n}{ \alpha_1})^k x_n)(like the last equation in the following figure.)