Entrance examination

examination for class entrance

1.

(10 minutes) Given is a residual $\bold r$ where

$$\bold r \quad = \quad \bold d_0 - m_1 \bold b_1 - m_2 \bold b_2 - m_3 \bold b_3$$

The data is $\bold d_0$.The fitting functions are the column vectors $\bold b_1$, $\bold b_2$, and $\bold b_3$,and the model parameters are the scalars m₁, m₂, and m₃. Suppose that m₁ and m₂ are already known. Derive a formula for finding m₃ that minimizes the residual length (squared) $\bold r \cdot \bold r$.

2.

(10 minutes) Below is a subroutine written in a mysterious dialect of Fortran. Describe ALL the inputs required for this subroutine to multiply a vector times the transpose of a matrix.

# matrix multiply and its adjoint
#
subroutine matmult( adj, bb,        x,nx,  y,ny)
integer ix, iy,     adj,              nx,    ny
real                     bb(ny,nx), x(nx), y(ny)
if( adj == 0 )
        do iy= 1, ny 
                y(iy) = 0.
else
        do ix= 1, nx 
                x(ix) = 0. 
do ix= 1, nx {
do iy= 1, ny {
        if( adj == 0 )
                        y(iy) = y(iy) + bb(iy,ix) * x(ix)
        else
                        x(ix) = x(ix) + bb(iy,ix) * y(iy)
        }}
return; end