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**Example text**

In fact, pTk Qpk+1 = = pTk Q(;gk+1 + k pk ) T Qp k T ;pTk Qgk+1 + gpk+1 T Qpk pk Qpk k = ;pTk Qgk+1 + gTk+1Qpk = 0 : It is somewhat more cumbersome to show that pi and pk+1 for i = 0 : : : k are also conjugate. This can be done by induction. 2) to produce conjugate rather than orthogonal vectors. Details can be found in Polak’s book mentioned earlier. 2 Removing the Hessian The algorithm shown in the previous subsection is a correct conjugate gradients algorithm. However, it is computationally inadequate because the expression for k contains the Hessian Q, which is too large.

Whenever two matrices A and B , diagonal or not, are related by A = QBQ;1 they are said to be similar to each other, and the transformation of transformation. 4) which is how eigenvalues and eigenvectors are usually introduced. In contrast, we have derived this equation from the requirement of diagonalizing a matrix by a similarity transformation. The columns of Q are called eigenvectors, and the diagonal entries of are called eigenvalues. 5) on a sample of points on the unit circle. The dashed lines are vectors that do not change direction under the transformation.

Bureau National Standards, section B, Vol 49, pp. 409-436, 1952), which also incorporates the steps from x0 to xn : g0 = g(x0 ) p0 = ;g0 for k = 0 : : : n;1 = arg min k 0 f(xk + pk ) xk+1 = xk + k pk gk+1 = g(xk+1) gT Qp k = pkTk+1Qpkk pk+1 = ;gk+1 + k pk end where gk = g(xk ) = @f @ x x=xk is the gradient of f at xk . It is simple to see that pk and pk+1 are conjugate. In fact, pTk Qpk+1 = = pTk Q(;gk+1 + k pk ) T Qp k T ;pTk Qgk+1 + gpk+1 T Qpk pk Qpk k = ;pTk Qgk+1 + gTk+1Qpk = 0 : It is somewhat more cumbersome to show that pi and pk+1 for i = 0 : : : k are also conjugate.