In this case L will not have 1s along the diagonal Algorithms used to compute

# In this case l will not have 1s along the diagonal

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In this case, L will not have 1’s along the diagonal. Algorithms used to compute Cholesky factorization are similar to those that compute LU factorizations. See the technical details at the end of these notes for further detail. The Cholesky decomposition is unique. It can actually be computed slightly faster than a general LU decomposition, and is easier to stabilize. Example. 6 3 0 3 4 1 0 1 3 = 2 . 4495 0 0 1 . 2247 1 . 5811 0 0 0 . 6325 1 . 6125 2 . 4495 1 . 2247 0 0 1 . 5811 0 . 6325 0 0 1 . 6125 6 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 13:22, November 13, 2019 Subscribe to view the full document.

QR factorization The QR decomposition factors A as A = QR , where Q is orthogonal, and R is upper triangular. It can be com- puted by running (a stabilized version of) Gram-Schmidt on the columns of A ; its computational complexity is again O ( N 3 ). Once we have it in hand, solving Ax = b has the cost of solving an or- thogonal system ( O ( N 2 )) and a triangular system ( O ( N 2 )). We will explore this connection more on the next homework. Like all of the other decompositions in this section, computing a QR decomposition costs O ( N 3 ), but once it is in place, we can solve Ax = b using x = R - 1 Q T b in O ( N 2 ) time. Symmetric QR When A is symmetric, we can write A = QT Q T , where Q is orthonormal, and T is symmetric and tri-diagonal : T = t 11 t 12 0 0 · · · 0 t 21 t 22 t 23 0 · · · 0 0 t 32 t 33 t 34 · · · 0 . . . . . . . . . 0 · · · 0 t NN - 1 t NN . 7 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 13:22, November 13, 2019 This is a handy fact, since in general, tridiagonal matrices are easier to manipulate (invert, compute eigenvalues/eigenvectors of, etc) that general symmetric matrices. We call this “symmetric QR ” since algorithms to compute this de- composition are very similar to those used to compute A = QR . See the technical details at the end of these notes for an example of such an algorithm. SVD and eigenvalue decompositions We are already familiar with the SVD for general matrices: A = U Σ V T . When A is square and invertible (rank( A ) = N ), then all of U , Σ , and V are N × N and UU T = U T U = V V T = V T V = I . As we have seen, we can solve Ax = b with x = V Σ - 1 U T b . We can see that with the SVD in place, the cost of solving a system is O ( N 2 ). For symmetric A , we can write A = V Λ V T , and then Ax = b is solved with x = V Λ - 1 V T b . Both of these decompositions represent a matrix as orthogonal-diagonal- orthogonal. Computing either costs O ( N 3 ), and they are slightly more expensive than the QR and LU decompositions above. In fact, computing a QR decomposition is often used as a stepping stone to computing the SVD or eigenvalue decomposition. 8 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 13:22, November 13, 2019 Subscribe to view the full document.

Computing eigenvalue decompositions of symmetric matrices For N × N symmetric positive semi-definite A , there are many ways to compute the eigenvalue decomposition A = V Λ V T . We discuss here one particular technique, popular for its stability, flexibility, and speed, based on power iterations.  • Fall '08
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