Implicit Q Theorem
Why Bulge Chasing = Explicit QR
Step 1 of 3
Initial Matrix H
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We start with an upper Hessenberg matrix H. We want to find its next state in the QR algorithm. We can do this explicitly (H -> R -> RQ) or implicitly (Bulge Chasing).
Explicit QR
0.50
-0.14
0.65
1.52
-0.23
-0.23
1.58
0.77
-0.47
0.54
-0.46
-0.47
0
-1.91
-1.72
-0.56
-1.01
0.31
0
0
1.47
-0.23
0.07
-1.42
0
0
0
0.38
-0.60
-0.29
0
0
0
0
0.82
-1.22
≈
Mathematically Equivalent
Implicit QR (Bulge Chasing)
0.50
-0.14
0.65
1.52
-0.23
-0.23
1.58
0.77
-0.47
0.54
-0.46
-0.47
0
-1.91
-1.72
-0.56
-1.01
0.31
0
0
1.47
-0.23
0.07
-1.42
0
0
0
0.38
-0.60
-0.29
0
0
0
0
0.82
-1.22