Gaussian elimination. System of Linear Equations. Gaussian elimination. System of Linear Equations

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Jord Cortadella Department of Computer Scence Introducton to Programmng Dept.

of equatons Fndng the nverse of a matrx Computng determnants Computng ranks and bases of

1 Jord Cortadella Department of Computer Scence Introducton to Programmng Dept. CS, UPC 2 An essental algorthm n Lnear Algebra wth multple applcatons: Solvng lnear systems of equatons Fndng the nverse of a matrx Computng determnants Computng ranks and bases of vector spaces Named after Carl Fredrch Gauss ( ), but known much before by the Chnese. Alan Turng contrbuted to provde modern numercal algorthms for computers (characterzaton of ll-condtoned systems). Introducton to Programmng Dept. CS, UPC 3 Introducton to Programmng Dept. CS, UPC 4

2 // Type defntons for real vectors and matrces usng rvector vector<double>; usng rmatrx vector<rvector>; // Pre: A s an n n matrx, b s an n-element vector. // Returns x such that Axb. In case A s sngular, // t returns a zero-szed vector. // Post: A and b are modfed. rvector SystemEquatons(rmatrx& A, rvector& b) { bool nvertble GaussElmnaton(A, b); f (not nvertble) return rvector(); // A s n row echelon form return BackSubsttuton(A, b); Axb A x b Assumpton: trangular system of equatons wthout zeroes n the dagonal Introducton to Programmng Dept. CS, UPC 5 Introducton to Programmng Dept. CS, UPC 6 Introducton to Programmng Dept. CS, UPC 7 Introducton to Programmng Dept. CS, UPC 8

3 // Pre: A s an nvertble n n matrx n row echelon form, // b s a n-element vector // Returns x such that Axb rvector BackSubsttuton(const rmatrx& A, const rvector& b) { nt n A.sze(); rvector x(n); // Creates the vector for the soluton // Calculates x from x[] to x[] for (nt n - 1; > ; --) { // The values x[+1..] have already been calculated double s ; for (nt j + 1; j < n; ++j) s s + A[][j] x[j]; x[] (b[] s)/a[][]; return x; k A b Invarant: The rows..k have been reduced (zeroes at the left of the dagonal) Introducton to Programmng Dept. CS, UPC 9 Introducton to Programmng Dept. CS, UPC 1 j k What s the best pvot? (max absolute value) swap rows For each row, add a multple of row k such that A[][k] becomes zero. The coeffcent s A[][k]/A[k][k]. Introducton to Programmng Dept. CS, UPC 11 Introducton to Programmng Dept. CS, UPC 12

// Pre: A s an n n matrx, b s a n-element vector. // Returns true f A s nvertble, and false f A s sngular. // Post: If A s nvertble, A and b are the result of // the (A s n row echelon form).

$// Returns the ndex of the row wth max absolute value // for the subcolumn A[k..][k]. nt fnd_max_pvot(const rmatrx& A, nt k) { nt n A.$ sze(); double max k; // ndex of the row wth max pvot double max_pvot abs(a[k][k]); swap(a[k], A[max]); swap(b[k], b[max]); // Swap rows k and max // Force s n column A[k+1.

sze(); double max k; // ndex of the row wth max pvot double max_pvot abs(a[k][k]); swap(a[k], A[max]); swap(b[k], b[max]); // Swap rows k and max // Force s n column A[k+1.

4 // Pre: A s an n n matrx, b s a n-element vector. // Returns true f A s nvertble, and false f A s sngular. // Post: If A s nvertble, A and b are the result of // the (A s n row echelon form). bool GaussanElmnaton(rmatrx& A, rvector& b) { nt n A.sze(); Dscusson: how large should a pvot be? // Reduce rows 1.. (use the pvot n prevous row k) for (nt k ; k < n - 1; ++k) { // Rows..k have already been reduced nt max fnd_max_pvot(a, k); // fnds the max pvot f (abs(a[max][k]) < 1e-1) return false; // Sngular matrx // Pre: A s an n n matrx, k s the ndex of a row. // Returns the ndex of the row wth max absolute value // for the subcolumn A[k..][k]. nt fnd_max_pvot(const rmatrx& A, nt k) { nt n A.sze(); double max k; // ndex of the row wth max pvot double max_pvot abs(a[k][k]); swap(a[k], A[max]); swap(b[k], b[max]); // Swap rows k and max // Force s n column A[k+1..][k] for (nt k + 1; < n; ++) { double c A[][k]/A[k][k]; // coeffcent to scale row A[][k] ; for (nt j k + 1; j < n; ++j) A[][j] A[][j] c A[k][j]; b[] b[] c b[k]; return true; // We have an nvertble matrx for (nt k + 1; < n; ++) { double a abs(a[][k]); f (a > max_pvot) { max_pvot a; max ; return max; Introducton to Programmng Dept. CS, UPC 13 Solvng multple systems of equatons Introducton to Programmng Dept. CS, UPC 14 // Pre: A s an nvertble n n matrx n row echelon form, // B s a n m matrx // Returns X such that AXB rmatrx BackSubsttuton(const rmatrx& A, const rmatrx& B) { nt n A.sze(); nt m B[].sze(); rmatrx X(n, rvector(m)); // Creates the soluton matrx // Calculates X from X[] to X[] for (nt n - 1; > ; --) { // The values X[+1..] have already been calculated for (nt k ; k < m; ++k) { double s ; for (nt j + 1; j < n; ++j) s s + A[][j] X[j][k]; X[][k] (B[][k] s)/a[][]; return X; Introducton to Programmng Dept. CS, UPC 15 Introducton to Programmng Dept. CS, UPC 16

5 Inverse of a Matrx // Pre: A s an n n matrx, B s an n m matrx. // Returns true f A s nvertble, and false f A s sngular. // Post: If A s nvertble, A and B are the result of // the (A s n row echelon form). bool GaussanElmnaton(rmatrx& A, rmatrx& B) { nt n A.sze(); nt m B[].sze(); Reduce the problem to a set of systems of lnear equatons: // Reduce rows 1.. (use the pvot n prevous row k) for (nt k ; k < n - 1; ++k) { // Rows..k have already been reduced nt max fnd_max_pvot(a, k); // fnds the max pvot f (abs(a[max][k]) < 1e-1) return false; // Sngular matrx swap(a[k], A[max]); swap(b[k], B[max]); // Swap rows k and max // Force s n column A[k+1..][k] for (nt k + 1; < n; ++) { double c A[][k]/A[k][k]; // coeffcent to scale row A[][k] ; for (nt j k + 1; j < n; ++j) A[][j] A[][j] c A[k][j]; for (nt l ; l < m; ++l) B[][l] B[][l] c B[k][l]; return true; // We have an nvertble matrx Introducton to Programmng Dept. CS, UPC 17 Computng determnants Rules of determnants: 1. Swappng two rows: determnant multpled by Multplyng a row by a scalar: the determnant s multpled by the same scalar 3. Addng to one row the scalar multple of another row: determnant does not change Algorthm: Do and remember the number of row swaps (odd or even). The determnant s the product of the elements n the dagonal (negated n case of an odd number of swaps). Rule 2 s not used, unless some row s scaled. Introducton to Programmng Dept. CS, UPC 18 Summary s the most used algorthm n Lnear Algebra. Complexty: O(n 3 ). There are many good packages to solve Lnear Algebra operatons (LAPACK, LINPACK, Matlab, Mathematca, NumPy, R, ). Introducton to Programmng Dept. CS, UPC 19 Introducton to Programmng Dept. CS, UPC 2

LU Decomposition Method Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America

LU Decomposition Method Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America nbm_sle_sm_ludecomp.nb 1 LU Decomposton Method Jame Trahan, Autar Kaw, Kevn Martn Unverst of South Florda Unted States of Amerca aw@eng.usf.edu nbm_sle_sm_ludecomp.nb 2 Introducton When solvng multple