Our unitary Hessenberg matrix A is now factored as a descending sequence of core transformations (n = 8)

1], and only B1, the bulge, disturbs the descending sequence of core transformations.

Exactly two of the core transformations from the descending sequence participate in the turnover.

We can use a turnover to pass a core transformation through a descending sequence as follows:

This leads in a simple manner to the QR-decomposition of the matrix Q in which we first perform an upgoing sequence of Givens transformations (removing the low rank part), followed by a descending sequence of Givens transformations (expanding the part of rank zero).

v1(1);u1]; Perform the descending sequence of Givens transformations for i=1:n1-1 M=[R1(i,i:n);u1(i+1)*v1(i),R1(i+1,i+1:n)]; [c,s,M] = Givensexp(M,v1(i:i+1)); b1(i:i+1)=[c,s;-conj(s),c]*b1(i:i+1); % Update the representation of the matrices R and u u1(i:i+1)=M(1:2,1).

Moreover, we can construct a special pattern (called an X-pattern), such that we start on top of the matrix with a descending sequence of rank expanding Givens transformations, and on the bottom with an upgoing rank decreasing sequence Givens transformations.

Based on these operations we can interchange the order of the upgoing and descending sequences of Givens transformations.

In case there are more upgoing and descending sequences of Givens transformations, one can also shift through all of the descending sequences.

Suppose we have a matrix brought to upper triangular form by performing two upgoing sequences of Givens transformations and two descending sequences of transformations (e.

And last week, when my eleven-year-old student was playing intermediate-level repertoire, I said, "With which motive in the next phrase does the

descending sequence begin that causes this phrase to decrescendo?