Approximation of L2 norm of matrix using gershgorin's circles

gershgorin.scala

object gershgorin extends App {
  type Matrix = Seq[Seq[Double]]
  def gershgorin(square: Matrix) = {
    val rows = (0 until square.size)
    rows.map(i => square(i)(i)) zip
    rows.map(i => (0 until i).map(j=> math.abs(square(i)(j))) ++ (i+1 until square.size).map(j=> math.abs(square(i)(j)))).map(_.sum)
  }
  def eigenValues(square: Matrix) = gershgorin(square).map{ cr => val (center,radius) = cr; (center - radius, center + radius)}
  def spectralRadius(square: Matrix) = eigenValues(square).map(_._2).max
  def mkString(m:Matrix) = m.map{r => r.mkString(",")}.mkString("\n")
  val symmetric = Seq(Seq(7,1,2.0), Seq(1,8,3.0), Seq(2,3,9.0))
  print("Matrix: \n%s\n Spectral Radius: %.2f \n eigenValues: %s".format( mkString(symmetric), spectralRadius(symmetric), eigenValues(symmetric)))
}

notes.txt

Given a nxn square matrix :
there exist n Gershgorin Circles = 1 circle per row
For each row, 
 center of circle = the diagonal cell. 
 radius of circle = sum the off-diagonal cell norms

Theorem: Eigen Values of a matrix must lie within its Gershgorin.
Each real eigen value is bounded by (center - radius, center + radius)

The Spectral Radius is the maximum eigen value.

For Symmetric Matrices:
The L2 norm of a symmetric marix is simply the spectral radius.
If the matrix ain't symmeric, all bets are off.

results.txt

$ scala gershgorin
Matrix: 
7.0,1.0,2.0
1.0,8.0,3.0
2.0,3.0,9.0
 Spectral Radius: 14.00 
 eigenValues: Vector((4.0,10.0), (4.0,12.0), (4.0,14.0))
 
From WolframAlpha:
 {12.4188, 6.38677, 5.1944}
 
 Notice that 
 12.4 lies between [4,14]
 6.4 lies between [4,12]
 5.2 lies between [4,10]
and true spectral radius = 12.4 is well approximated by 14

For a symmetric matrix, L2 norm = spectral radius.