2.1 Gaussian Elimination
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1 2. Gaussian Elimination A common problem encountered in numerical models is the one in which there are n equations and n unknowns. The following is a description of the Gaussian elimination method for an nxn system of linear equations, where A is the coefficient matrix, x is the unknown vector, and d is the given right hand side vector. Ax= dwhere, A is nxn. For the special case that the system consists of only one linear equation, the coefficient matrix reduces to a single entry [a], and the solution can be easily determined by division. following form The system of linear equations is usually expressed as an augmented matrix of the [ A d ]. We will use elementary row operations to transform coefficient matrix, A, into another matrix, U, which is an upper triangular matrix. The augmented matrix is said to have been triangularized when [ A d ] U d ˆ. Three row operations exist for performing this transformation. They are. interchanging rows, 2. replace a row by a nonzero multiple of itself and 3. replace a row by a combination of itself plus a multiple of another row.
2 2 The transformation is attained by multiplying both the coefficient and right hand side portions of the augmented matrix by E, a product of elementary matrices that correspond to the row operations. Note the definitions for U and ˆd. ( Π E) [ A d] = [( ΠE) A ( Π Ed ) ] = U d Then ( Π EA ) = U and ( Π Ed ) = d. Now solve the triangularized system of linear equations by a backward sweep beginning with the last equation. Using the solution obtained from the last equation, the next-to-last equation can be solved. This process is repeated until the first equation is reached, whereby the system will have been completely solved. Ux= d u x = d nn, n n u x + u x = d n, n n n, n n n It should be noted that when dealing with floating point arithmetic, errors will be incurred if u n-,n- is, or is near to, zero. Example. We wish to perform an LU factorization on the following linear system by Gaussian elimination = [ A d] 3 0 2
3 3 First, we want to place a 0 in place of the 2 in row 2 and column. To do this, subtract 2 times the first row from the second row, i.e. R2-2(R) E2[ A d], E = = E = We define E 2, E 3, E 32 to be elementary matrices that correspond to the row operations E3E2[ A d], E = = = E E E [ A d], E = Therefore, we have shown that the original augmented matrix can be multiplied by three elementary matrices to form an upper triangular matrix. Note that these elementary matrices are lower triangular in form. [ ] ˆ U d = E32E3E2 A d. Multiply both sides by the inverses of E 2, E 3, E 32 to obtain the following [ ] E ˆ 2E3E 32 U d = A d E E E U = A.
4 4 Multiplying the three elementary matrices results in a 3x3 lower triangular matrix, which is the lower (L) factorization portion of the LU-factorization of the matrix A = A An LU-factorization has thus been found for the matrix, A. Definition. If L and U are lower and upper triangular matrices and A = LU, then LU is called an LU factorization of A. Consider the following linear system Ax= d. Suppose A has an LU factorization LU = A. Define y by the following equation, LUx ( ) = d and Ux= y. This results in the following set of systems, Ly = d and Ux= y. First, we will solve Ly = d, and then we can solve Ux = y.
5 5 Consider the stage k in a forward sweep of the Gaussian elimination method where A = aij a a a a d 0 a a d 0 0 a a d () () () () () 2 3 n (2) (2) (2) (2) 0 a22 a23 a2n d2 (3) (3) (3) 0 0 a33 a3n d3 ( k) ( k) ( k) kk kn k ( k) ( n) ( n) nk nn n We wish to make all entries, in column k below the diagonal, a nk (k) equal to zero in the matrix, so as to produce an upper triangular matrix, U. This will be accomplished by applying the following row operation, multiplication by an elementary matrix, a ( row i) m ( row k) where m =, i > k E ( k) ( k) akk = m
6 6 Proposition. If each diagonal entry ( or pivot), a kk (k), is nonzero, then A = LU. Proof: The assumption implies that each E exists. Thus, a 0 Π E A= U ( k) kk ^ ^ ( ). A= ΠE U where Π denotes reverse order Therefore, E A = U. Since each E - exists, A = LU. Gaussian Elimination Algorithm with Overwriting of A for the LU factorization (kij version, or row version) for k = :n- for i = k+:n m = a /a kk a = m for j = k+:n a ij = a ij - m a kj
7 7 The number of floating-point operations for this algorithm is O(2n 3 /3). Furthermore, with appropriated modifications the order of the loops may be changed to give 3! = 6 possible variations. One important variations is the kji version, or column version.
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