[Assignment 4] Part 2, finding K and b

[yichangC]

Hi Prof. and TA,

I'm trying to find the K and b for the lease-squares error in quadratic from:
S^T K s + S^T b + c

The way to solve it is explained in this issue:
danielepanozzo/gp#13

My questions are:

1. In the issue, it's mentioned that "the "G" matrix I believe you refer to, which can be computed by igl::grad, holds the G_t for each triangle t (but not directly stacked: it chooses to place row i of G_t in G.row(i * #F + t))"
   I don't really understand what it means, could you provide more explanation?

2. According to @jpanetta, the expression could be written as: E_1(s) = s^T G^T G s - 2 s^T G^T u + const, however, G^T (3F * V) * u (F * 3) cannot produce a valid matrix. Am I missing anything?

Thanks.

[jiangzhongshi]

Hi,

1. The gradient of a scalar function on the mesh (G*f) will have three components for each triangle, say (Gf)_x, (Gf)_y, (Gf)_z, each has size |F| * |V| and is then stacked together. Each row i of G_t corresponds to a coordinate (x/y/z), so the stack rule you wrote, corresponds to stacking, vertically, (Gf)_x, (Gf)_y, (Gf)_z together.

2. After understanding what G means, the three columns of u, is just the x/y/z component, and you would like to put them in correspondence of three components of G. Therefore, just flatten u into a (3F1) vector will suffice. Also, G is 3FV, so G^T should be V*3F

[LihengGong]

Hi Zhongshi,
Quick questions:

1. By stacking, vertically, (Gf)_x, (Gf)_y, (Gf)_z together. do you mean that all x coordinates are stacked together(forming the first #F rows), followed by all y coordinates(forming the second #F rows), then followed by all z coordinates(forming the last #F rows)?

2. By the three columns of u, is just the x/y/z component, you would like to put them in correspondence of three components of G do you mean that we stack matrix u into a vector such that the first #F rows are the first column, the second #F rows are the second column and the last #F rows are the third column?

Thank you!