Update README.md

README.md

@@ -6,6 +6,9 @@ Update the "Q-less" QR factorization of a matrix. Routines are
 provided for adding and deleting columns, adding rows, and solving
 associated linear least-squares problems.
 
+The least-squares solver uses Björck's corrected semi-normal equation (CSNE) approach [1] with one step of iterative refinement. Using double precision, the method should be stable for matrices `A` with condition number up to `10^8`.
+
+
 ## Installing
 
 ```julia
@@ -16,7 +19,7 @@ Pkg.add(url="https://github.com/mpf/QRupdate.jl")
 
 ### Adding columns
 
-Build the "Q-less" QR factorization of A one column at a time.
+Build the "Q-less" QR factorization of `A` one column at a time:
 
 ```julia
 m, n = 100, 50
@@ -29,7 +32,7 @@ for i in 1:n
 end
 ```
 
-Solve a least-squares problem using R.
+Minimize the least-squares residual `||Ax - b||₂` using the computed `R`:
 
 ```julia
 b = randn(m)
@@ -38,7 +41,7 @@ x, r = csne(R, A, b)
 
 ### Deleting columns
 
-Delete a column and compute new R.
+Delete a random column and compute new `R`:
 
 ```julia
 n = size(A,2)
@@ -47,15 +50,15 @@ A = A[:, 1:n .!= k]
 R = qrdelcol(R, k)
 ```
 
-Solve a least-squares problem using R.
+Minimize the least-squares residual `||Ax - b||₂` using the computed `R`:
 
 ```julia
 x, r = csne(R, A, b)
 ```
 
 ### Adding rows
 
-Add a row to A.
+Add a row to `A`:
 
 ```julia
 n = size(A,2)
@@ -64,7 +67,7 @@ A = [A; a]
 R = qraddrow(R, a)
 ```
 
-Solve a least-squares problem using R.
+Minimize the least-squares residual `||Ax - b||₂` using the computed `R`:
 
 ```julia
 b = [b; randn()]
@@ -73,15 +76,14 @@ x, r = csne(R, A, b)
 
 ## Reference
 
-Björck, A. (1996). Numerical methods for least squares problems. SIAM.
+[1] Björck, A. (1996). Numerical methods for least squares problems. SIAM.
 
 ## Change Log
 
-- 15 Jun 2007: First version of QRaddcol.m (without β).
-    - Where necessary, Ake Bjorck's CSNE (Corrected Semi-Normal Equations) method is used to improve the accuracy of `u` and `γ`.  See p143 of Ake Bjork's Least Squares book.
-
+- 15 Jun 2007: First version of QRaddcol.m (without `β`).
+  - Where necessary, Ake Bjorck's CSNE method is used to improve the accuracy of `u` and `γ`.  See p143 of Ake Bjork's Least Squares book.
 - 18 Jun 2007: `R` is now the exact size on entry and exit.
 - 19 Oct 2007: Sparse `A`, a makes `c` and `u` sparse. Force them to be dense.
 - 04 Aug 2008: Update `u` using `du`, rather than `u = R*z` as in Ake's book.  We guess that it might be slightly more accurate, but it's hard to tell.  No `R*z` makes it a little cheaper.
 - 03 Sep 2008: Generalize `A` to be `[A; β*I]` for some scalar `β`.  Update `u` using `du`, but keep Ake's version in comments.
-- 29 Dec 2015: Converted to Julia.
+- 29 Dec 2015: Converted to Julia.