#RFC: Updated spectralnorm.jl for ~30x speedup

The previous version based on Scala ran in just under 5 minutes (~293s), while this version,
based on Javascript with tweaks, runs consistently in 10-12s. This puts Julia ahead of all
other dynamic languages and only behind C/Ada/Fortran using extremely low-level and in some
cases parallelization (which maybe we should explore). Julia FTW!
Let me know if we can make any other tweaks/optimizations. 

examples/shootout/spectralnorm.jl

@@ -3,43 +3,45 @@
 # spectral-norm benchmark
 # https://shootout.alioth.debian.org/u32/performance.php?test=spectralnorm
 #
-# Based on the Scala program
-#
+# Based on the Javascript program
+# 
+
+include("timing.jl")
+
+A(i,j) = 1.0 / ((i+j)*(i+j+1.0)/2.0+i+1.0)
 
-# infinite matrix function and its transpose
-A(i, j) = 1.0 / ((i+j)*(i+j+1)/2 +i+1)
-At(j, i) = 1.0 / ((i+j)*(i+j+1)/2 +i+1)
-
-# w <- M * v
-function mult(N, v, w, M)
- for i = 1:N
- s = 0.0
- for j = 1:N
- s += M(i-1, j-1) * v[j]
- end
- w[i] = s
- end
+function Au(u,w)
+ n = length(u)
+ for i = 1:n, j = 1:n
+ j == 1 && (w[i] = 0)
+ w[i] += A(i-1,j-1) * u[j]
+ end
 end
 
-function approximate(N)
- u = ones(N)
- v = ones(N)
- w = ones(N)
- for i = 1:10
- mult(N, u, w, A)
- mult(N, w, v, At)
- mult(N, v, w, A)
- mult(N, w, u, At)
- end
-
- vbv = 0.0
- vv = 0.0
- for i = 1:N
- vbv += u[i] * v[i]
- vv += v[i] * v[i]
- end
-
- sqrt(vbv / vv)
+function Atu(w,v)
+ n = length(w)
+ for i = 1:n, j = 1:n
+ j == 1 && (v[i] = 0)
+ v[i] += A(j-1,i-1) * w[j]
+ end
+end
+
+function approximate(n)
+ u = ones(Float64,n)
+ v = zeros(Float64,n)
+ w = zeros(Float64,n)
+ vv = vBv = 0
+ for i = 1:10
+ Au(u,w)
+ Atu(w,v)
+ Au(v,w)
+ Atu(w,u)
+ end
+ for i = 1:n
+ vBv += u[i]*v[i]
+ vv += v[i]*v[i]
+ end
+ return sqrt(vBv/vv)
 end
 
 function spectralnorm(N)
@@ -52,3 +54,8 @@ else
  N = 100
 end
 spectralnorm(N)
+
+# @assert spectralnorm(100) == 1.274219991
+# @timeit spectralnorm(500) "spectralnorm(n=500)"
+# @timeit spectralnorm(3000) "spectralnorm(n=3000)"
+# @timeit spectralnorm(5500) "spectralnorm(n=5500)"