SlideShare a Scribd company logo

Factorization from Gaussian Elimination

•Download as PPT, PDF•
•
Mudassir Parvi
Mudassir Parvi

Its a topic Of Numerical Computing contains Matlab code aswell

Education
1 of 49
분산시스템 연구실 6. LU Factorization 12-cs-029 Mudassir Ali
분산시스템 연구실 Contents • LU Factorization from Gaussian Elimination • LU Factorization of Tridiagonal Matrices • LU Factorization with Pivoting • Direct LU Factorization • Application of LU Factorization • Matlab’s Method
분산시스템 연구실 Example. Circuit Analysis Application  Solve by LU Factorization of coefficient matrix 0V1 = 0V2 = 200V3 = 20R1 = 25R3 = 30R5 =10R2 = 10R4 = 1i 2i 3i 200)(10)(1030 0)(20)(1025 0)(10)(20 13233 12322 3121 =-+-+ =-+-+ =-+- iiiii Flow around lower loop iiiii Flow around upper loop iiii Flow around the left loop
분산시스템 연구실 6.1 LU Factorization from Gaussian Elimination
분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.1           = 28143 1062 321 A  Example 6.1 Three-By-Three System           = 100 010 001 L, ,           = 28143 1062 321 U Step 1:           = 103 012 001 L ,           = 1980 420 321 U Step 2:           = 103 012 001 L ,           = 1980 420 321 U Multiply L by U, and Verify the result: LU =           103 012 001           1980 420 321 . =           28143 1062 321 = A           = 28143 1062 321 A ,           = 28143 1062 321 A ,
분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.2  Example 6.2 Four-By-Four System             = 1415113 202092 91871 48124 A Step 1: Row 1 is unchanged, and rows 2-4 are modified to give 4 1 11 21 21 == a a l 2 1 11 31 31 == a a l 4 3 11 41 31 == a a l             = 11920 181630 81640 48124 U             = 1000 0100 0010 0001 L             = 1415113 202092 91871 48124 U             = 1415113 202092 91871 48124 A             = 11500 0100 0014/1 0001 L             −−− = 1415113 202092 19218370 48124 U             = 1415113 202092 91871 48124 A             = 11500 0102/1 0014/1 0001 L             = 1415113 181630 81640 48124 U             = 1415113 202092 91871 48124 A             = 11504/3 0102/1 0014/1 0001 L
분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.2 Step 2: Row 1 and 2 are unchanged, row 3 and 4 are transformed, yielding Example 6.2 Four-By-Four System 4 3 22 32 32 == a a l 2 1 22 42 42 == a a l             = 7100 12400 81640 48124 U Step 3: The fourth row is modified to complete the forward elimination stage: 4 1 33 43 43 == a a l             = 4000 12400 81640 48124 U             = 102/14/3 014/32/1 0014/1 0001 L             = 1415113 202092 91871 48124 A             = 14/12/14/3 014/32/1 0014/1 0001 L             = 1415113 202092 91871 48124 A
분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.2 Multiply L by U to verify the result: Example 6.2 Four-By-Four System                   1 4 1 2 1 4 3 01 4 3 2 1 001 4 1 0001             4000 12400 81640 48124 .             1415113 202092 91871 48124 = L . U = A
분산시스템 연구실 LU Factorization from Gaussian Elimination - MATLAB Function for LU Factorization 6.1.1 MATLAB Function for LU Factorization Using Gaussian Elimination function [L, U] = LU_Factor(A) % LU factorization of matrix A % using Gaussian elimination without pivoting % Input : A  n-by-n matrix % Output L ( lower triangular ) and % U (upper triangular ) [n, m] = size(A); L = eye(n); % initialize matrices U = A; for j = 1 : n for I = j + 1 : n L( i , j ) = U( i, j ) / U( j, j ); U( i , : ) = U( i, j ) / L( j, j ) * U( j, : ); end end % display L and U L U % verify results B = L * U A
분산시스템 연구실 LU Factorization from Gaussian Elimination - MATLAB Function for LU Factorization >> LU_factor(A); L = 1.0000 0 0 0 0.2500 1.0000 0 0 0 0 1.0000 0 0 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0 1.0000 0 0 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0 1.0000 0 0.7500 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0.5000 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0.5000 0.2500 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0.5000 0.2500 1.0000 U = 4 12 8 4 0 4 16 8 0 0 4 12 0 0 0 4 B = 4 12 8 4 1 7 18 9 2 9 20 20 3 11 15 14 A = 4 12 8 4 1 7 18 9 2 9 20 20 3 11 15 14
분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.3  Example 6.3 LU Factorization for Circuit Analysis Examp 1321 102030 Viii =−− 2321 105520 Viii =−+− 3321 551010 Viii =+−−           −− −− −− = 501010 105520 102030 A Step 1: The matrices after the first stage of Gaussian elimination are           − −= 103/1 013/2 001 L           − − −− = 3/1403/500 3/503/1250 102030 U, Step 2: The matrices after the second stage of Gaussian elimination are           −− −= 15/23/1 013/2 001 L           − −− = 4000 3/503/1250 102030 U,
분산시스템 연구실 LU Factorization from Gaussian Elimination - Discussion  6.1.2 Discussion           100 01 001 c .           333231 232221 131211 aaa aaa aaa =           333231 232221 131211 aaa ddd aaa Here, d12 = ca11+a21, d22=ca12+a22, and d32 = ca13+a32. We write this as CA = D We also need the inverse of the matrix C           − 100 01 001 c .           100 01 001 c =           100 010 001 , or C-1 . C = I .
분산시스템 연구실 LU Factorization from Gaussian Elimination - Discussion           − − 10 01 001 32 21 m m .           10 01 001 31 21 m m =           100 010 001 M1 -1 . M1 = I .           − 10 010 001 32m .           10 010 001 32m =           100 010 001 M2 -1 . M2 = I .           − − 10 01 001 32 21 m m           − 10 010 001 32m . =           −− − 1 01 001 3232 21 mm m M1 -1 . M2 -1 = I .
분산시스템 연구실 6.2 LU Factorization of Tridiagonal Matrix
분산시스템 연구실 LU Factorization of Tridiagonal Matrix • 6.2.1 Matlab function for LU Factorization of a Tridiagonal Matrix  LU Factorization of a tridiagonal matrix T • less computation • less computer memory
분산시스템 연구실 LU Factorization of Tridiagonal Matrix function [dd, bb] = LU_tridiag(a, d, b) % LU factorization of a tridiagonal matrix T % Input % a vector of elements above main diagonal, a(n) = 0 % d diagonal of matrix T % b vector of elements below main diagonal, b(1) = 0 % The factorization of T consists of % Lower bidiagonal matrix, % 1’s on main diagonal; lower diagonal is bb % Upper bidiagonal matrix, % main diagonal is dd; upper diagonal is a N = length(d) bb(1) = 0; dd(1) = d(1); for i = 2 : n bb(i) = b(i) / dd(i-1); dd(i) = d(i) – bb(i) * a(i-1); end
분산시스템 연구실 LU Factorization of Tridiagonal Matrix Example 6.4 Consider again the four-by-four tridigonal matrix from Example 3.7,             − −− −− − = 2100 1210 0121 0012 M four-by-four tridigonal matrix             = 44 333 222 11 00 0 0 00 db adb adb ad T  Example 6.4 LU Factorization of Tridigonal System             = 2 2 2 2 d             − − − = 0 1 1 1 a             − − − = 1 1 1 0 b, , ,             − −− −− − = 2100 1210 0121 0012 M             = 100 010 001 0001 4 3 2 bb bb bb L             = 4 33 22 11 000 00 00 00 dd add add add U
분산시스템 연구실 LU Factorization of Tridiagonal Matrix Example 6.4 1) For i = 2, dd1 = d1 = 2, a1 = -1 bb2 = b2/dd1 = -1/2 dd2 = d2 – bb2a1 = 2-(-1/2)(-1) = 3/2             − = 100 010 0012/1 0001 4 3 bb bb L             − = 4 33 2 000 00 02/30 0012 dd add a U 2) For i = 3, a2 = -1 bb3 = b3/dd2 = -1/(3/2) = -2/3 dd3 = d3 - bb3a2 = 2-(-2/3)(-1) = 4/3             − − = 100 013/20 0012/1 0001 4bb L             − − = 4 3 000 3/400 012/30 0012 dd a U
분산시스템 연구실 LU Factorization of Tridiagonal Matrix Example 6.4 3) For i = 4, a3 = -1 bb4 = b4/dd2 = -1/(4/3) = -3/4 dd4 = d4 - bb4a3 = 2-(-3/4)(-1) = 5/4             − − − = 14/300 013/20 0012/1 0001 L             − − − = 4/5000 13/400 012/30 0012 U
분산시스템 연구실 6.3 LU Factorization with pivoting
분산시스템 연구실 6.3.1 Why pivoting?           =           − ⋅           ⋅=⋅           −=           =           = 331 28143 1062 200 1350 1062 102/1 012/3 001 ???? 1350 200 1062 ... 102/3 012/1 001 28143 331 1062 APUL UL A
분산시스템 연구실 6.3.2 Theory for pivoting 1/2 • 한 쪽에 A 행렬 , 다른 한쪽에 I 행렬을 놓는 다 . (A: LU 분해 할 NxN 행렬 , I: NxN 단위행 렬 ) • A 행렬을 가우스 소거법을 이용해 상삼각 행 렬로 변환하면서 A 행렬에 가하는 행렬 연산 을 기록한다 . (Nx: 소거 , Px: 교환 ) ULAP UNNNAPPP UAPPPNNN UAPNPNPN nnn nn nn ⋅=⋅ ⋅⋅⋅⋅=⋅⋅⋅⋅ =⋅⋅⋅⋅⋅⋅⋅⋅ =⋅⋅⋅⋅⋅⋅⋅ − − − − − − −− −− 1 1 1 2 1 1121 121121 112211 ''' '''   
분산시스템 연구실 6.3.2 Theory for pivoting 2/2 • Nx 행렬 을 좌측으 로 빼면서 N’x 로 변 환한다 . • 좌측 N’x 들의 역행 렬을 I 행 렬에 곱한 다 .             =             − − −             •             − − − =             − − − •             − 1002 0103 0014 0001 1002 0103 0014 0001 0010 0100 1000 0001 1002 0103 0014 0001 1004 0103 0012 0001 0010 0100 1000 0001 1
분산시스템 연구실 6.3.3 pivoting example 1/2           =           ⋅           ⋅           − −⋅                     =           ⋅           ⋅           − −           =           ⋅                     =           220 040 888 888 442 484 001 010 100 102/1 014/1 001 010 100 001 040 220 888 888 442 484 001 010 100 102/1 014/1 001 484 442 888 888 442 484 001 010 100 888 442 484 888 442 484
분산시스템 연구실 6.3.3 pivoting example 2/2           ⋅           =           ⋅                     ⋅           ⋅           =           ⋅           ⋅                     =           ⋅           ⋅           ⋅           − −⋅           −           =           ⋅           ⋅           − −⋅           ⋅           − 200 040 888 12/12/1 014/1 001 888 442 484 010 001 100 200 040 888 12/10 010 001 102/1 014/1 001 888 442 484 001 010 100 010 100 001 200 040 888 888 442 484 001 010 100 010 100 001 102/1 014/1 001 12/10 010 001 200 040 888 888 442 484 001 010 100 104/1 012/1 001 010 100 001 12/10 010 001
분산시스템 연구실 6.3.4 pivoting procedure                                                                                                                                                 0010 0100 0001 1000 2220 3100 0040 8888 1008/1 0102/1 0014/1 0001 0001 0100 0010 1000 0040 3100 2220 8888 1002/1 0108/1 0014/1 0001 0001 0100 0010 1000 4484 4211 4442 8888 1000 0100 0010 0001 1000 0100 0010 0001 8888 4211 4442 4484 1000 0100 0010 0001                                                                                                                 0100 0010 0001 1000 2000 2200 0040 8888 12/102/1 012/18/1 0014/1 0001 0100 0010 0001 1000 3100 2200 0040 8888 1002/1 012/18/1 0014/1 0001 0010 0100 0001 1000 2200 3100 0040 8888 102/18/1 0102/1 0014/1 0001   
분산시스템 연구실 6.3.5 pivoting code Function [L, U, P] = LU_pivot(A) [n, n1] = size(A); L=eye(n); P=eye(n); U=A; For j = 1:n [pivot m] = max(abs(U(j:n, j))); m = m+j-1; if m ~= j % interchange rows m and j in U % interchange rows m and j in P if j >= 2 % interchange rows m and j in columns 1:j-1 of L end end for i = j+1:n L(i, j) = U(i, j) / U(j, j); U(i, :) = U(i, :) – L(i, j)*U(j, :); end end
분산시스템 연구실 6.4 Direct LU Factorization
분산시스템 연구실 6.4.1 Theory for Direct LU factorization )( )( 00 0 0 00 1111 1111 2211 21 22221 11211 222121111 222121222212211121 11112111111 21 22221 11211 222 11211 21 2221 11 jiululula jiululula ululule aaa aaa aaa eululul ululululul ululul aaa aaa aaa u uu uuu lll ll l jjijjjjijiij ijiijiiijiij nnnnnnnnnn nnnn n n nnnnn nn n nnnn n n nn n n nnnn ≥⋅+⋅++⋅= ≤⋅+⋅++⋅= ⋅++⋅+⋅=             =             ⋅+⋅⋅ ⋅+⋅⋅+⋅⋅ ⋅⋅⋅             =             •             −− −−                    
분산시스템 연구실 6.4.2 Type of Direct LU factorization • 조건의 수 = N^2 변수의 수 = N(N+1)/2 + N(N+1)/2 = N^2+N • Doolittle 방식 L 의 대각 원소가 1 • Crout 방식 U 의 대각 원소가 1 • Cholesky 방식 L 과 U 의 같은 위치 대각 원소끼리 같 음
분산시스템 연구실 6.4.3 Doolittle example           =           •                     =           •                     =           •                     =           •           64325 32204 541 300 1240 541 135 014 001 64325 32204 541 00 1240 541 135 014 001 64325 32204 541 00 0 541 15 014 001 64325 32204 541 00 0 1 01 001 33 33 2322 32 33 2322 131211 3231 21 u u uu l u uu uuu ll l
분산시스템 연구실 6.4.4 Doolittle code Function [L, U] = Doolittle(A) [n, m] = size(A); U = zeros(n, m); L = eye(n); for k = 1:n U(k, k) = A(k, k) – L(k, 1:k-1)*U(1:k-1, k); for j = k+1:n U(k, j) = A(k, j) – L(k, 1:k-1)*U(1:k-1, j); L(j,k)= (A(j,k)– L(j, 1:k-1)*U(1:k-1, k))/U(k,k); end end
분산시스템 연구실 6.4.5 Cholesky example           =           •                     =           •                     =           •                     =           •           64325 32204 541 300 620 541 365 024 001 64325 32204 541 00 620 541 65 024 001 64325 32204 541 00 0 541 5 04 001 64325 32204 541 00 00 00 3333 33 2322 3332 22 33 2322 131211 333231 2221 11 xx x ux xl x x ux uux xll xl x
분산시스템 연구실 6.4.6 Cholesky code Function [L, U] = Cholesky(A) % A is assumed to be symmetric. % L is computed, and U = L’ [n, m] = size(A); L = zeros(n, n); for k = 1:n L(k, k) = sqrt( A(k,k) – L(k, 1:k-1)*L(k, 1:k-1)’ ); for i = k+1:n L(i, k) = (A(i,k) – L(i, 1:k-1)*L(k, 1:k-1)’)/L(k, k); end end
분산시스템 연구실 6.5 Applications of LU Factorization
분산시스템 연구실 Contents • 6.5 Applications of LU Factorization – Linear Equations – Tridiagonal System – Determinant of a Matrix – Inverse of a Matrix • 6.6 MATLAB’s Methods – lu, chol, det, inv
분산시스템 연구실 6.5.1 Solving Systems of Linear Equations • Ax = b → LUx = b → Ly = b – A = LU, Ux = y • Solve the system Ly = b for y – Forward substitution – y1 → y2 → …… yn • Solve the system Ux = y for x – Backward substitution – xn → xn-1 → …… x1 • If pivoting has been used – Ax=b → PAx=Pb, where PA=LU, Pb=c
분산시스템 연구실 32)80)(5/2(,80,0onsubstitutiforwardBy 0 80 0 15/23/1 013/2 001 4000 50/3-125/30 10-20-30 15/23/1 013/2 001 0 80 0 501010 105520 102030 321 3 2 1 ====           =                     −− −⇒=           =           −− −=⇒=           =           −− −− −− =⇒= yyy y y y bLy ULLUA bAbAx Ex 6.8 Solving Electrical Circuit for Several Voltages
분산시스템 연구실 Ex 6.8 Solving Electrical Circuit for Several Voltages 11 22 33 3 2 1 0.8044/2510(4/5)]-(56/25)(1/30)[-20 2.2456/255)40/3)(3/12(80 0.804/532/40 onsubstitutibackwardBy 32 80 0 4000 3/503/1250 1020-30 ix ix ix x x x ←=== ←==+= ←===           =                     − − ⇒= yUx
분산시스템 연구실 MATLAB Function to Solve the Linear System LUx=b function x = LU_Solve(L, U, b) % Function to solve the equation L U x = b % L --> Lower triangular matrix (with 1's on diagonal) % U --> Upper triangular matrix % b --> Right-hand side vector [n m] = size(L); z = zeros(n,1); x = zeros(n,1); % Solve L z = b using forward substitution z(1) = b(1); for i = 2:n z(i) = b(i) - L(i, 1:i-1) * z(1:i-1); end % Solve U x = z using back substitution x(n) = z(n) / U(n, n); for i = n-1 : -1 : 1 x(i) = (z(i) - U(i,i+1:n) * x(i+1:n)) / U(i, i); end
분산시스템 연구실 6.5.2 Solving a Tridiagonal System [ ] [ ] [ ] [ ] [ ] [ ]54-32-1 r)bb,dd,_solve(a,LU_tridiag 1111-143240 2173151 4324005214 89 6 5 23 7 214000 57300 02320 001154 00041 = = =−= ⇒= ==                 − − =                 −=⇒= x x ddb d bba rArAx Ex 6.9
분산시스템 연구실 MATLAB Function for Solving a Tridiagonal System function x = LU_tridiag_solve(a, d, b, r) % Function to solve the eqquation A x = r % LU factorization of A is expressed as % Lower bidiagonal matrix: % diagonal is 1s, lower diagonal is b; b(1) = 0. % Upper bidiagonal matrix: diagonal is d, upper diagonal is a % Right-hand-side vector is r % Solve L z = r using forward substitution n = length(d); z(1) = r(1); for i = 2:n z(i) = r(i) - b(i) * z(i-1); end % Solve U x = z using back substitution x(n) = z(n) / d(n); for i = n-1 : -1 : 1 x(i) = (z(i) - a(i) * x(i+1)) / d(i); end
분산시스템 연구실 6.5.3 Determinant of a Matrix ionfactorizatLUtheoccurredthat esinterchangrowofnumbertheiswhere )1()det( thatsoused,ispivotingIf )det(then,If 11 11 k ul ul n i ii n i ii k n i ii n i ii ∏∏ ∏∏ == == −= = == A LUPA ALUA -1 8)8)(1)(1()det( 800 510 211 111 012 001 121 112 211 332211 −=−==           − − =           −− −=⇒           − − − = uuuA ULA Ex 6.10
분산시스템 연구실 6.5.4 Inverse of a Matrix • The inverse of an n-by-n matrix A Axi=ei (i=1, … ,n) – ei=[0 0 … 1 … 0 0]’, where the 1 appears in the ith position – X whose columns are the solution vectors x1, … , xn is A-1
분산시스템 연구실 Ex 6.11 Finding a Matrix Inverse Using LU Factorization           =⇒           =                     −− −           =                     −− − =           =           −− −=           = 113 012 001 0 0 1 111 012 001 ofcolumningcorrespondon,substitutiforward:ofcolumnEach 100 010 001 111 012 001 forSolve 800 51-0 21-1 111 012 001 121- 112- 21-1 31 21 11 333231 232221 131211 Y IY YILY ULA y y y yyy yyy yyy
분산시스템 연구실 Ex 6.11 Finding a Matrix Inverse Using LU Factorization           −− − ==           =                     − − == − 8/18/18/3 8/58/38/1 8/38/58/1 ofcolumningcorrespondon,substitutiback:XofcolumnEach 113 012 001 800 510 211 then;forSolve 1 333231 232221 131211 1 AX Y XAXYUX xxx xxx xxx -
분산시스템 연구실 MATLAB Function function x = LU_Solve_Gen(L, U, B) % Function to solve the equation L U x = B % L --> Lower triangular matrix (1's on diagonal) % U --> Upper triangular matrix % B --> Right-hand-side matrix [n n2] = size(L); [m1 m] = size(B); % Solve L z = B using forward substitution for j = 1:m z(1, j) = b(1, j); for i = 2 : n z(i,j) = B(i,j) - L(i, 1:i-1) * z(1:i-1, j); end end % Solve U x = z using back substitution for j = 1:m x(n, j) = z(n, j) / U(n, n); for i = n-1 : -1 : 1 x(i,j) = (z(i,j)-U(i,i+1:n)*x(i+1:n,j)) / U(i,i); end end
분산시스템 연구실 6.6 MATLAB’s Method
분산시스템 연구실 6.6 MATLAB’s Method • 4 built-in functions – LU decomposition of a square matrix A • [L, U] = lu(A), [L, U, P] = lu(A) – Cholesky factorization • chol – Determinant of a matrix • det – Inverse of a matrix • inv

Recommended

Lu decomposition
Lu decompositionLu decomposition
Lu decompositiongilandio
 
Lu Decomposition
Lu DecompositionLu Decomposition
Lu DecompositionMdAlAmin187
 
Solving linear equation system
Solving linear equation systemSolving linear equation system
Solving linear equation systemsulaiman_karim
 
Maths
MathsMaths
MathsVignesh Ravichandran
 
Modelling Planar Vehicle Dynamics using Bicycle Model
Modelling Planar Vehicle Dynamics using Bicycle ModelModelling Planar Vehicle Dynamics using Bicycle Model
Modelling Planar Vehicle Dynamics using Bicycle ModelAkshay Mistri
 
Es272 ch4b
Es272 ch4bEs272 ch4b
Es272 ch4bBatuhan Yıldırım
 
Newton Forward Difference Interpolation Method
Newton Forward Difference Interpolation MethodNewton Forward Difference Interpolation Method
Newton Forward Difference Interpolation MethodAdeel Rasheed
 
B.Tech-II_Unit-V
B.Tech-II_Unit-VB.Tech-II_Unit-V
B.Tech-II_Unit-VKundan Kumar
 

Recommended

Lu decomposition
Lu decompositionLu decomposition
Lu decompositiongilandio
 
Lu Decomposition
Lu DecompositionLu Decomposition
Lu DecompositionMdAlAmin187
 
Solving linear equation system
Solving linear equation systemSolving linear equation system
Solving linear equation systemsulaiman_karim
 
Maths
MathsMaths
MathsVignesh Ravichandran
 
Modelling Planar Vehicle Dynamics using Bicycle Model
Modelling Planar Vehicle Dynamics using Bicycle ModelModelling Planar Vehicle Dynamics using Bicycle Model
Modelling Planar Vehicle Dynamics using Bicycle ModelAkshay Mistri
 
Es272 ch4b
Es272 ch4bEs272 ch4b
Es272 ch4bBatuhan Yıldırım
 
Newton Forward Difference Interpolation Method
Newton Forward Difference Interpolation MethodNewton Forward Difference Interpolation Method
Newton Forward Difference Interpolation MethodAdeel Rasheed
 
B.Tech-II_Unit-V
B.Tech-II_Unit-VB.Tech-II_Unit-V
B.Tech-II_Unit-VKundan Kumar
 
Eigenvalues and eigenvectors
Eigenvalues and eigenvectorsEigenvalues and eigenvectors
Eigenvalues and eigenvectorsiraq
 
Multiple intigration ppt
Multiple intigration pptMultiple intigration ppt
Multiple intigration pptManish Mor
 
First order ordinary differential equations and applications
First order ordinary differential equations and applicationsFirst order ordinary differential equations and applications
First order ordinary differential equations and applicationsAdani Institute of Infrastructure Engineering College
 
Gauss y gauss jordan
Gauss y gauss jordanGauss y gauss jordan
Gauss y gauss jordanjonathann89
 
Jacobi method for MATLAB
Jacobi method for MATLAB Jacobi method for MATLAB
Jacobi method for MATLAB Taimoor Muzaffar Gondal
 
the fourier series
the fourier seriesthe fourier series
the fourier seriessafi al amu
 
6.4 inverse matrices t
6.4 inverse matrices t6.4 inverse matrices t
6.4 inverse matrices tmath260
 
Gaussian Elimination Method
Gaussian Elimination MethodGaussian Elimination Method
Gaussian Elimination MethodAndi Firdaus
 
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...Smit Shah
 
Gauss elimination
Gauss eliminationGauss elimination
Gauss eliminationluiscarlosmolina
 
Jacobi method
Jacobi methodJacobi method
Jacobi methodGrishma Maravia
 
Interpolation
InterpolationInterpolation
InterpolationBrijesh Padhiyar
 
Euler's Method
Euler's MethodEuler's Method
Euler's Methoddmidgette
 
Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
 
Application of Cylindrical and Spherical coordinate system in double-triple i...
Application of Cylindrical and Spherical coordinate system in double-triple i...Application of Cylindrical and Spherical coordinate system in double-triple i...
Application of Cylindrical and Spherical coordinate system in double-triple i...Sonendra Kumar Gupta
 
Line integral & ML inequality
Line integral & ML inequalityLine integral & ML inequality
Line integral & ML inequalitymalharrana101
 
Mb 106 quantitative techniques 2
Mb 106 quantitative techniques 2Mb 106 quantitative techniques 2
Mb 106 quantitative techniques 2KrishnaRoy45
 
Odepowerpointpresentation1
Odepowerpointpresentation1 Odepowerpointpresentation1
Odepowerpointpresentation1 Pokarn Narkhede
 
System of linear equations and their solution
System of linear equations and their solutionSystem of linear equations and their solution
System of linear equations and their solutionJoseph Nilo
 
Vector space - subspace By Jatin Dhola
Vector space - subspace By Jatin DholaVector space - subspace By Jatin Dhola
Vector space - subspace By Jatin DholaJatin Dhola
 
Computer Architecture Performance Evolution of Caches using Patch
Computer Architecture Performance Evolution of Caches using Patch Computer Architecture Performance Evolution of Caches using Patch
Computer Architecture Performance Evolution of Caches using Patch Mudassir Parvi
 
Choleskymethod
CholeskymethodCholeskymethod
Choleskymethoduis
 

More Related Content

What's hot

Eigenvalues and eigenvectors
Eigenvalues and eigenvectorsEigenvalues and eigenvectors
Eigenvalues and eigenvectorsiraq
 
Multiple intigration ppt
Multiple intigration pptMultiple intigration ppt
Multiple intigration pptManish Mor
 
Gauss y gauss jordan
Gauss y gauss jordanGauss y gauss jordan
Gauss y gauss jordanjonathann89
 
the fourier series
the fourier seriesthe fourier series
the fourier seriessafi al amu
 
6.4 inverse matrices t
6.4 inverse matrices t6.4 inverse matrices t
6.4 inverse matrices tmath260
 
Gaussian Elimination Method
Gaussian Elimination MethodGaussian Elimination Method
Gaussian Elimination MethodAndi Firdaus
 
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...Smit Shah
 
Euler's Method
Euler's MethodEuler's Method
Euler's Methoddmidgette
 
Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
 
Application of Cylindrical and Spherical coordinate system in double-triple i...
Application of Cylindrical and Spherical coordinate system in double-triple i...Application of Cylindrical and Spherical coordinate system in double-triple i...
Application of Cylindrical and Spherical coordinate system in double-triple i...Sonendra Kumar Gupta
 
Line integral & ML inequality
Line integral & ML inequalityLine integral & ML inequality
Line integral & ML inequalitymalharrana101
 
Mb 106 quantitative techniques 2
Mb 106 quantitative techniques 2Mb 106 quantitative techniques 2
Mb 106 quantitative techniques 2KrishnaRoy45
 
Odepowerpointpresentation1
Odepowerpointpresentation1 Odepowerpointpresentation1
Odepowerpointpresentation1 Pokarn Narkhede
 
System of linear equations and their solution
System of linear equations and their solutionSystem of linear equations and their solution
System of linear equations and their solutionJoseph Nilo
 
Vector space - subspace By Jatin Dhola
Vector space - subspace By Jatin DholaVector space - subspace By Jatin Dhola
Vector space - subspace By Jatin DholaJatin Dhola
 

What's hot (20)

Eigenvalues and eigenvectors
Eigenvalues and eigenvectorsEigenvalues and eigenvectors
Eigenvalues and eigenvectors
 
Multiple intigration ppt
Multiple intigration pptMultiple intigration ppt
Multiple intigration ppt
 
First order ordinary differential equations and applications
First order ordinary differential equations and applicationsFirst order ordinary differential equations and applications
First order ordinary differential equations and applications
 
Gauss y gauss jordan
Gauss y gauss jordanGauss y gauss jordan
Gauss y gauss jordan
 
Jacobi method for MATLAB
Jacobi method for MATLAB Jacobi method for MATLAB
Jacobi method for MATLAB
 
the fourier series
the fourier seriesthe fourier series
the fourier series
 
6.4 inverse matrices t
6.4 inverse matrices t6.4 inverse matrices t
6.4 inverse matrices t
 
Gaussian Elimination Method
Gaussian Elimination MethodGaussian Elimination Method
Gaussian Elimination Method
 
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
 
Gauss elimination
Gauss eliminationGauss elimination
Gauss elimination
 
Jacobi method
Jacobi methodJacobi method
Jacobi method
 
Interpolation
InterpolationInterpolation
Interpolation
 
Euler's Method
Euler's MethodEuler's Method
Euler's Method
 
Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )Cauchy integral theorem & formula (complex variable & numerical method )
Cauchy integral theorem & formula (complex variable & numerical method )
 
Application of Cylindrical and Spherical coordinate system in double-triple i...
Application of Cylindrical and Spherical coordinate system in double-triple i...Application of Cylindrical and Spherical coordinate system in double-triple i...
Application of Cylindrical and Spherical coordinate system in double-triple i...
 
Line integral & ML inequality
Line integral & ML inequalityLine integral & ML inequality
Line integral & ML inequality
 
Mb 106 quantitative techniques 2
Mb 106 quantitative techniques 2Mb 106 quantitative techniques 2
Mb 106 quantitative techniques 2
 
Odepowerpointpresentation1
Odepowerpointpresentation1 Odepowerpointpresentation1
Odepowerpointpresentation1
 
System of linear equations and their solution
System of linear equations and their solutionSystem of linear equations and their solution
System of linear equations and their solution
 
Vector space - subspace By Jatin Dhola
Vector space - subspace By Jatin DholaVector space - subspace By Jatin Dhola
Vector space - subspace By Jatin Dhola
 

Viewers also liked

Computer Architecture Performance Evolution of Caches using Patch
Computer Architecture Performance Evolution of Caches using Patch Computer Architecture Performance Evolution of Caches using Patch
Computer Architecture Performance Evolution of Caches using Patch Mudassir Parvi
 
Choleskymethod
CholeskymethodCholeskymethod
Choleskymethoduis
 
Matrix factorization
Matrix factorizationMatrix factorization
Matrix factorizationrubyyc
 
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2njit-ronbrown
 
Specials Methods
Specials MethodsSpecials Methods
Specials MethodsUIS
 
Cholesky method and Thomas
Cholesky method and ThomasCholesky method and Thomas
Cholesky method and Thomasjorgeduardooo
 
Chapter 4: Linear Algebraic Equations
Chapter 4: Linear Algebraic EquationsChapter 4: Linear Algebraic Equations
Chapter 4: Linear Algebraic EquationsMaria Fernanda
 
Crout s method for solving system of linear equations
Crout s method for solving system of linear equationsCrout s method for solving system of linear equations
Crout s method for solving system of linear equationsSugathan Velloth
 
Presentation of project on e auction with dotnet
Presentation of project on e auction with dotnetPresentation of project on e auction with dotnet
Presentation of project on e auction with dotnetSunanda Chakraborty
 
Tính toán khoa học - Chương 2: Hệ phương trình tuyến tính
Tính toán khoa học - Chương 2: Hệ phương trình tuyến tínhTính toán khoa học - Chương 2: Hệ phương trình tuyến tính
Tính toán khoa học - Chương 2: Hệ phương trình tuyến tínhChien Dang
 
Gauss Elimination Method With Partial Pivoting
Gauss Elimination Method With Partial PivotingGauss Elimination Method With Partial Pivoting
Gauss Elimination Method With Partial PivotingSM. Aurnob
 
Matlab practical and lab session
Matlab practical and lab sessionMatlab practical and lab session
Matlab practical and lab sessionDr. Krishna Mohbey
 
SRS on online auction system
SRS on online auction systemSRS on online auction system
SRS on online auction systemsagar_paperwala
 

Viewers also liked (20)

Computer Architecture Performance Evolution of Caches using Patch
Computer Architecture Performance Evolution of Caches using Patch Computer Architecture Performance Evolution of Caches using Patch
Computer Architecture Performance Evolution of Caches using Patch
 
Choleskymethod
CholeskymethodCholeskymethod
Choleskymethod
 
Matrix factorization
Matrix factorizationMatrix factorization
Matrix factorization
 
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
 
Specials Methods
Specials MethodsSpecials Methods
Specials Methods
 
Lu decomposition
Lu decompositionLu decomposition
Lu decomposition
 
Linear and non linear equation
Linear and non linear equationLinear and non linear equation
Linear and non linear equation
 
Cholesky method and Thomas
Cholesky method and ThomasCholesky method and Thomas
Cholesky method and Thomas
 
Unit5
Unit5Unit5
Unit5
 
Chapter 4: Linear Algebraic Equations
Chapter 4: Linear Algebraic EquationsChapter 4: Linear Algebraic Equations
Chapter 4: Linear Algebraic Equations
 
Crout s method for solving system of linear equations
Crout s method for solving system of linear equationsCrout s method for solving system of linear equations
Crout s method for solving system of linear equations
 
Gauss seidel
Gauss seidelGauss seidel
Gauss seidel
 
Presentation of project on e auction with dotnet
Presentation of project on e auction with dotnetPresentation of project on e auction with dotnet
Presentation of project on e auction with dotnet
 
Tính toán khoa học - Chương 2: Hệ phương trình tuyến tính
Tính toán khoa học - Chương 2: Hệ phương trình tuyến tínhTính toán khoa học - Chương 2: Hệ phương trình tuyến tính
Tính toán khoa học - Chương 2: Hệ phương trình tuyến tính
 
Gauss Elimination Method With Partial Pivoting
Gauss Elimination Method With Partial PivotingGauss Elimination Method With Partial Pivoting
Gauss Elimination Method With Partial Pivoting
 
Matlab programs
Matlab programsMatlab programs
Matlab programs
 
Matlab practical and lab session
Matlab practical and lab sessionMatlab practical and lab session
Matlab practical and lab session
 
Gauss sediel
Gauss sedielGauss sediel
Gauss sediel
 
Auction presentation
Auction presentationAuction presentation
Auction presentation
 
SRS on online auction system
SRS on online auction systemSRS on online auction system
SRS on online auction system
 

Similar to Factorization from Gaussian Elimination

Numerical Methods: Solution of system of equations
Numerical Methods: Solution of system of equationsNumerical Methods: Solution of system of equations
Numerical Methods: Solution of system of equationsNikolai Priezjev
 
07. LU Decomposition.pptx
07. LU Decomposition.pptx07. LU Decomposition.pptx
07. LU Decomposition.pptxkibriaswe
 
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...SAJJAD KHUDHUR ABBAS
 
Solutions_Manual_to_accompany_Applied_Nu.pdf
Solutions_Manual_to_accompany_Applied_Nu.pdfSolutions_Manual_to_accompany_Applied_Nu.pdf
Solutions_Manual_to_accompany_Applied_Nu.pdfWaleedHussain30
 
Econometric Analysis 8th Edition Greene Solutions Manual
Econometric Analysis 8th Edition Greene Solutions ManualEconometric Analysis 8th Edition Greene Solutions Manual
Econometric Analysis 8th Edition Greene Solutions ManualLewisSimmonss
 
18-21 Principles of Least Squares.ppt
18-21 Principles of Least Squares.ppt18-21 Principles of Least Squares.ppt
18-21 Principles of Least Squares.pptBAGARAGAZAROMUALD2
 
block diagram representation of control systems
block diagram representation of control systemsblock diagram representation of control systems
block diagram representation of control systemsAhmed Elmorsy
 
Lecture (3) _transportation problem31-10-2020.pdf
Lecture (3) _transportation problem31-10-2020.pdfLecture (3) _transportation problem31-10-2020.pdf
Lecture (3) _transportation problem31-10-2020.pdfamrhebafamily
 
Consider the system.docx
Consider the system.docxConsider the system.docx
Consider the system.docxhoneyarguelles
 
Chapter 20 solutions
Chapter 20 solutionsChapter 20 solutions
Chapter 20 solutionsLK Education
 
Solution of System of Linear Equations
Solution of System of Linear EquationsSolution of System of Linear Equations
Solution of System of Linear Equationsmofassair
 
MODI_SteppingStone.pptx
MODI_SteppingStone.pptxMODI_SteppingStone.pptx
MODI_SteppingStone.pptxVivekSaurabh7
 

Similar to Factorization from Gaussian Elimination (20)

Numerical Methods: Solution of system of equations
Numerical Methods: Solution of system of equationsNumerical Methods: Solution of system of equations
Numerical Methods: Solution of system of equations
 
07. LU Decomposition.pptx
07. LU Decomposition.pptx07. LU Decomposition.pptx
07. LU Decomposition.pptx
 
Lelt 240 semestre i-2021
Lelt 240 semestre i-2021Lelt 240 semestre i-2021
Lelt 240 semestre i-2021
 
Ejerciciooo3
Ejerciciooo3Ejerciciooo3
Ejerciciooo3
 
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...
 
Solutions_Manual_to_accompany_Applied_Nu.pdf
Solutions_Manual_to_accompany_Applied_Nu.pdfSolutions_Manual_to_accompany_Applied_Nu.pdf
Solutions_Manual_to_accompany_Applied_Nu.pdf
 
Ab data
Ab dataAb data
Ab data
 
Central tedancy & correlation project - 2
Central tedancy & correlation project - 2Central tedancy & correlation project - 2
Central tedancy & correlation project - 2
 
Capítulo 03 materiais
Capítulo 03 materiaisCapítulo 03 materiais
Capítulo 03 materiais
 
Econometric Analysis 8th Edition Greene Solutions Manual
Econometric Analysis 8th Edition Greene Solutions ManualEconometric Analysis 8th Edition Greene Solutions Manual
Econometric Analysis 8th Edition Greene Solutions Manual
 
Matlab
MatlabMatlab
Matlab
 
18-21 Principles of Least Squares.ppt
18-21 Principles of Least Squares.ppt18-21 Principles of Least Squares.ppt
18-21 Principles of Least Squares.ppt
 
block diagram representation of control systems
block diagram representation of control systemsblock diagram representation of control systems
block diagram representation of control systems
 
Lecture (3) _transportation problem31-10-2020.pdf
Lecture (3) _transportation problem31-10-2020.pdfLecture (3) _transportation problem31-10-2020.pdf
Lecture (3) _transportation problem31-10-2020.pdf
 
Consider the system.docx
Consider the system.docxConsider the system.docx
Consider the system.docx
 
Chapter 20 solutions
Chapter 20 solutionsChapter 20 solutions
Chapter 20 solutions
 
Solution of System of Linear Equations
Solution of System of Linear EquationsSolution of System of Linear Equations
Solution of System of Linear Equations
 
MODI_SteppingStone.pptx
MODI_SteppingStone.pptxMODI_SteppingStone.pptx
MODI_SteppingStone.pptx
 
MODI
MODIMODI
MODI
 
Static Models of Continuous Variables
Static Models of Continuous VariablesStatic Models of Continuous Variables
Static Models of Continuous Variables
 

Recently uploaded

Call Girls in Dwarka Mor Delhi Contact Us 9654467111
Call Girls in Dwarka Mor Delhi Contact Us 9654467111Call Girls in Dwarka Mor Delhi Contact Us 9654467111
Call Girls in Dwarka Mor Delhi Contact Us 9654467111Sapana Sha
 
18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf
18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf
18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdfssuser54595a
 
Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdfEnzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdfSumit Tiwari
 
_Math 4-Q4 Week 5.pptx Steps in Collecting Data
_Math 4-Q4 Week 5.pptx Steps in Collecting Data_Math 4-Q4 Week 5.pptx Steps in Collecting Data
_Math 4-Q4 Week 5.pptx Steps in Collecting DataJhengPantaleon
 
The Most Excellent Way | 1 Corinthians 13
The Most Excellent Way | 1 Corinthians 13The Most Excellent Way | 1 Corinthians 13
The Most Excellent Way | 1 Corinthians 13Steve Thomason
 
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...EduSkills OECD
 
A Critique of the Proposed National Education Policy Reform
A Critique of the Proposed National Education Policy ReformA Critique of the Proposed National Education Policy Reform
A Critique of the Proposed National Education Policy ReformChameera Dedduwage
 
Contemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptx
Contemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptxContemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptx
Contemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptxRoyAbrique
 
How to Configure Email Server in Odoo 17
How to Configure Email Server in Odoo 17How to Configure Email Server in Odoo 17
How to Configure Email Server in Odoo 17Celine George
 
Paris 2024 Olympic Geographies - an activity
Paris 2024 Olympic Geographies - an activityParis 2024 Olympic Geographies - an activity
Paris 2024 Olympic Geographies - an activityGeoBlogs
 
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...Marc Dusseiller Dusjagr
 
CARE OF CHILD IN INCUBATOR..........pptx
CARE OF CHILD IN INCUBATOR..........pptxCARE OF CHILD IN INCUBATOR..........pptx
CARE OF CHILD IN INCUBATOR..........pptxGaneshChakor2
 
The basics of sentences session 2pptx copy.pptx
The basics of sentences session 2pptx copy.pptxThe basics of sentences session 2pptx copy.pptx
The basics of sentences session 2pptx copy.pptxheathfieldcps1
 
Class 11 Legal Studies Ch-1 Concept of State .pdf
Class 11 Legal Studies Ch-1 Concept of State .pdfClass 11 Legal Studies Ch-1 Concept of State .pdf
Class 11 Legal Studies Ch-1 Concept of State .pdfakmcokerachita
 
microwave assisted reaction. General introduction
microwave assisted reaction. General introductionmicrowave assisted reaction. General introduction
microwave assisted reaction. General introductionMaksud Ahmed
 
SOCIAL AND HISTORICAL CONTEXT - LFTVD.pptx
SOCIAL AND HISTORICAL CONTEXT - LFTVD.pptxSOCIAL AND HISTORICAL CONTEXT - LFTVD.pptx
SOCIAL AND HISTORICAL CONTEXT - LFTVD.pptxiammrhaywood
 

Recently uploaded (20)

Call Girls in Dwarka Mor Delhi Contact Us 9654467111
Call Girls in Dwarka Mor Delhi Contact Us 9654467111Call Girls in Dwarka Mor Delhi Contact Us 9654467111
Call Girls in Dwarka Mor Delhi Contact Us 9654467111
 
18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf
18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf
18-04-UA_REPORT_MEDIALITERAСY_INDEX-DM_23-1-final-eng.pdf
 
Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdfEnzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
 
_Math 4-Q4 Week 5.pptx Steps in Collecting Data
_Math 4-Q4 Week 5.pptx Steps in Collecting Data_Math 4-Q4 Week 5.pptx Steps in Collecting Data
_Math 4-Q4 Week 5.pptx Steps in Collecting Data
 
The Most Excellent Way | 1 Corinthians 13
The Most Excellent Way | 1 Corinthians 13The Most Excellent Way | 1 Corinthians 13
The Most Excellent Way | 1 Corinthians 13
 
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
 
TataKelola dan KamSiber Kecerdasan Buatan v022.pdf
TataKelola dan KamSiber Kecerdasan Buatan v022.pdfTataKelola dan KamSiber Kecerdasan Buatan v022.pdf
TataKelola dan KamSiber Kecerdasan Buatan v022.pdf
 
9953330565 Low Rate Call Girls In Rohini Delhi NCR
9953330565 Low Rate Call Girls In Rohini Delhi NCR9953330565 Low Rate Call Girls In Rohini Delhi NCR
9953330565 Low Rate Call Girls In Rohini Delhi NCR
 
A Critique of the Proposed National Education Policy Reform
A Critique of the Proposed National Education Policy ReformA Critique of the Proposed National Education Policy Reform
A Critique of the Proposed National Education Policy Reform
 
Contemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptx
Contemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptxContemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptx
Contemporary philippine arts from the regions_PPT_Module_12 [Autosaved] (1).pptx
 
How to Configure Email Server in Odoo 17
How to Configure Email Server in Odoo 17How to Configure Email Server in Odoo 17
How to Configure Email Server in Odoo 17
 
Paris 2024 Olympic Geographies - an activity
Paris 2024 Olympic Geographies - an activityParis 2024 Olympic Geographies - an activity
Paris 2024 Olympic Geographies - an activity
 
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...
 
CARE OF CHILD IN INCUBATOR..........pptx
CARE OF CHILD IN INCUBATOR..........pptxCARE OF CHILD IN INCUBATOR..........pptx
CARE OF CHILD IN INCUBATOR..........pptx
 
Model Call Girl in Bikash Puri Delhi reach out to us at 🔝9953056974🔝
Model Call Girl in Bikash Puri Delhi reach out to us at 🔝9953056974🔝Model Call Girl in Bikash Puri Delhi reach out to us at 🔝9953056974🔝
Model Call Girl in Bikash Puri Delhi reach out to us at 🔝9953056974🔝
 
The basics of sentences session 2pptx copy.pptx
The basics of sentences session 2pptx copy.pptxThe basics of sentences session 2pptx copy.pptx
The basics of sentences session 2pptx copy.pptx
 
Class 11 Legal Studies Ch-1 Concept of State .pdf
Class 11 Legal Studies Ch-1 Concept of State .pdfClass 11 Legal Studies Ch-1 Concept of State .pdf
Class 11 Legal Studies Ch-1 Concept of State .pdf
 
microwave assisted reaction. General introduction
microwave assisted reaction. General introductionmicrowave assisted reaction. General introduction
microwave assisted reaction. General introduction
 
Staff of Color (SOC) Retention Efforts DDSD
Staff of Color (SOC) Retention Efforts DDSDStaff of Color (SOC) Retention Efforts DDSD
Staff of Color (SOC) Retention Efforts DDSD
 
SOCIAL AND HISTORICAL CONTEXT - LFTVD.pptx
SOCIAL AND HISTORICAL CONTEXT - LFTVD.pptxSOCIAL AND HISTORICAL CONTEXT - LFTVD.pptx
SOCIAL AND HISTORICAL CONTEXT - LFTVD.pptx
 

Factorization from Gaussian Elimination

  • 1. 분산시스템 연구실 6. LU Factorization 12-cs-029 Mudassir Ali
  • 2. 분산시스템 연구실 Contents • LU Factorization from Gaussian Elimination • LU Factorization of Tridiagonal Matrices • LU Factorization with Pivoting • Direct LU Factorization • Application of LU Factorization • Matlab’s Method
  • 3. 분산시스템 연구실 Example. Circuit Analysis Application  Solve by LU Factorization of coefficient matrix 0V1 = 0V2 = 200V3 = 20R1 = 25R3 = 30R5 =10R2 = 10R4 = 1i 2i 3i 200)(10)(1030 0)(20)(1025 0)(10)(20 13233 12322 3121 =-+-+ =-+-+ =-+- iiiii Flow around lower loop iiiii Flow around upper loop iiii Flow around the left loop
  • 4. 분산시스템 연구실 6.1 LU Factorization from Gaussian Elimination
  • 5. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.1           = 28143 1062 321 A  Example 6.1 Three-By-Three System           = 100 010 001 L, ,           = 28143 1062 321 U Step 1:           = 103 012 001 L ,           = 1980 420 321 U Step 2:           = 103 012 001 L ,           = 1980 420 321 U Multiply L by U, and Verify the result: LU =           103 012 001           1980 420 321 . =           28143 1062 321 = A           = 28143 1062 321 A ,           = 28143 1062 321 A ,
  • 6. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.2  Example 6.2 Four-By-Four System             = 1415113 202092 91871 48124 A Step 1: Row 1 is unchanged, and rows 2-4 are modified to give 4 1 11 21 21 == a a l 2 1 11 31 31 == a a l 4 3 11 41 31 == a a l             = 11920 181630 81640 48124 U             = 1000 0100 0010 0001 L             = 1415113 202092 91871 48124 U             = 1415113 202092 91871 48124 A             = 11500 0100 0014/1 0001 L             −−− = 1415113 202092 19218370 48124 U             = 1415113 202092 91871 48124 A             = 11500 0102/1 0014/1 0001 L             = 1415113 181630 81640 48124 U             = 1415113 202092 91871 48124 A             = 11504/3 0102/1 0014/1 0001 L
  • 7. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.2 Step 2: Row 1 and 2 are unchanged, row 3 and 4 are transformed, yielding Example 6.2 Four-By-Four System 4 3 22 32 32 == a a l 2 1 22 42 42 == a a l             = 7100 12400 81640 48124 U Step 3: The fourth row is modified to complete the forward elimination stage: 4 1 33 43 43 == a a l             = 4000 12400 81640 48124 U             = 102/14/3 014/32/1 0014/1 0001 L             = 1415113 202092 91871 48124 A             = 14/12/14/3 014/32/1 0014/1 0001 L             = 1415113 202092 91871 48124 A
  • 8. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.2 Multiply L by U to verify the result: Example 6.2 Four-By-Four System                   1 4 1 2 1 4 3 01 4 3 2 1 001 4 1 0001             4000 12400 81640 48124 .             1415113 202092 91871 48124 = L . U = A
  • 9. 분산시스템 연구실 LU Factorization from Gaussian Elimination - MATLAB Function for LU Factorization 6.1.1 MATLAB Function for LU Factorization Using Gaussian Elimination function [L, U] = LU_Factor(A) % LU factorization of matrix A % using Gaussian elimination without pivoting % Input : A  n-by-n matrix % Output L ( lower triangular ) and % U (upper triangular ) [n, m] = size(A); L = eye(n); % initialize matrices U = A; for j = 1 : n for I = j + 1 : n L( i , j ) = U( i, j ) / U( j, j ); U( i , : ) = U( i, j ) / L( j, j ) * U( j, : ); end end % display L and U L U % verify results B = L * U A
  • 10. 분산시스템 연구실 LU Factorization from Gaussian Elimination - MATLAB Function for LU Factorization >> LU_factor(A); L = 1.0000 0 0 0 0.2500 1.0000 0 0 0 0 1.0000 0 0 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0 1.0000 0 0 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0 1.0000 0 0.7500 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0.5000 0 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0.5000 0.2500 1.0000 L = 1.0000 0 0 0 0.2500 1.0000 0 0 0.5000 0.7500 1.0000 0 0.7500 0.5000 0.2500 1.0000 U = 4 12 8 4 0 4 16 8 0 0 4 12 0 0 0 4 B = 4 12 8 4 1 7 18 9 2 9 20 20 3 11 15 14 A = 4 12 8 4 1 7 18 9 2 9 20 20 3 11 15 14
  • 11. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Example 6.3  Example 6.3 LU Factorization for Circuit Analysis Examp 1321 102030 Viii =−− 2321 105520 Viii =−+− 3321 551010 Viii =+−−           −− −− −− = 501010 105520 102030 A Step 1: The matrices after the first stage of Gaussian elimination are           − −= 103/1 013/2 001 L           − − −− = 3/1403/500 3/503/1250 102030 U, Step 2: The matrices after the second stage of Gaussian elimination are           −− −= 15/23/1 013/2 001 L           − −− = 4000 3/503/1250 102030 U,
  • 12. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Discussion  6.1.2 Discussion           100 01 001 c .           333231 232221 131211 aaa aaa aaa =           333231 232221 131211 aaa ddd aaa Here, d12 = ca11+a21, d22=ca12+a22, and d32 = ca13+a32. We write this as CA = D We also need the inverse of the matrix C           − 100 01 001 c .           100 01 001 c =           100 010 001 , or C-1 . C = I .
  • 13. 분산시스템 연구실 LU Factorization from Gaussian Elimination - Discussion           − − 10 01 001 32 21 m m .           10 01 001 31 21 m m =           100 010 001 M1 -1 . M1 = I .           − 10 010 001 32m .           10 010 001 32m =           100 010 001 M2 -1 . M2 = I .           − − 10 01 001 32 21 m m           − 10 010 001 32m . =           −− − 1 01 001 3232 21 mm m M1 -1 . M2 -1 = I .
  • 14. 분산시스템 연구실 6.2 LU Factorization of Tridiagonal Matrix
  • 15. 분산시스템 연구실 LU Factorization of Tridiagonal Matrix • 6.2.1 Matlab function for LU Factorization of a Tridiagonal Matrix  LU Factorization of a tridiagonal matrix T • less computation • less computer memory
  • 16. 분산시스템 연구실 LU Factorization of Tridiagonal Matrix function [dd, bb] = LU_tridiag(a, d, b) % LU factorization of a tridiagonal matrix T % Input % a vector of elements above main diagonal, a(n) = 0 % d diagonal of matrix T % b vector of elements below main diagonal, b(1) = 0 % The factorization of T consists of % Lower bidiagonal matrix, % 1’s on main diagonal; lower diagonal is bb % Upper bidiagonal matrix, % main diagonal is dd; upper diagonal is a N = length(d) bb(1) = 0; dd(1) = d(1); for i = 2 : n bb(i) = b(i) / dd(i-1); dd(i) = d(i) – bb(i) * a(i-1); end
  • 17. 분산시스템 연구실 LU Factorization of Tridiagonal Matrix Example 6.4 Consider again the four-by-four tridigonal matrix from Example 3.7,             − −− −− − = 2100 1210 0121 0012 M four-by-four tridigonal matrix             = 44 333 222 11 00 0 0 00 db adb adb ad T  Example 6.4 LU Factorization of Tridigonal System             = 2 2 2 2 d             − − − = 0 1 1 1 a             − − − = 1 1 1 0 b, , ,             − −− −− − = 2100 1210 0121 0012 M             = 100 010 001 0001 4 3 2 bb bb bb L             = 4 33 22 11 000 00 00 00 dd add add add U
  • 18. 분산시스템 연구실 LU Factorization of Tridiagonal Matrix Example 6.4 1) For i = 2, dd1 = d1 = 2, a1 = -1 bb2 = b2/dd1 = -1/2 dd2 = d2 – bb2a1 = 2-(-1/2)(-1) = 3/2             − = 100 010 0012/1 0001 4 3 bb bb L             − = 4 33 2 000 00 02/30 0012 dd add a U 2) For i = 3, a2 = -1 bb3 = b3/dd2 = -1/(3/2) = -2/3 dd3 = d3 - bb3a2 = 2-(-2/3)(-1) = 4/3             − − = 100 013/20 0012/1 0001 4bb L             − − = 4 3 000 3/400 012/30 0012 dd a U
  • 19. 분산시스템 연구실 LU Factorization of Tridiagonal Matrix Example 6.4 3) For i = 4, a3 = -1 bb4 = b4/dd2 = -1/(4/3) = -3/4 dd4 = d4 - bb4a3 = 2-(-3/4)(-1) = 5/4             − − − = 14/300 013/20 0012/1 0001 L             − − − = 4/5000 13/400 012/30 0012 U
  • 20. 분산시스템 연구실 6.3 LU Factorization with pivoting
  • 21. 분산시스템 연구실 6.3.1 Why pivoting?           =           − ⋅           ⋅=⋅           −=           =           = 331 28143 1062 200 1350 1062 102/1 012/3 001 ???? 1350 200 1062 ... 102/3 012/1 001 28143 331 1062 APUL UL A
  • 22. 분산시스템 연구실 6.3.2 Theory for pivoting 1/2 • 한 쪽에 A 행렬 , 다른 한쪽에 I 행렬을 놓는 다 . (A: LU 분해 할 NxN 행렬 , I: NxN 단위행 렬 ) • A 행렬을 가우스 소거법을 이용해 상삼각 행 렬로 변환하면서 A 행렬에 가하는 행렬 연산 을 기록한다 . (Nx: 소거 , Px: 교환 ) ULAP UNNNAPPP UAPPPNNN UAPNPNPN nnn nn nn ⋅=⋅ ⋅⋅⋅⋅=⋅⋅⋅⋅ =⋅⋅⋅⋅⋅⋅⋅⋅ =⋅⋅⋅⋅⋅⋅⋅ − − − − − − −− −− 1 1 1 2 1 1121 121121 112211 ''' '''   
  • 23. 분산시스템 연구실 6.3.2 Theory for pivoting 2/2 • Nx 행렬 을 좌측으 로 빼면서 N’x 로 변 환한다 . • 좌측 N’x 들의 역행 렬을 I 행 렬에 곱한 다 .             =             − − −             •             − − − =             − − − •             − 1002 0103 0014 0001 1002 0103 0014 0001 0010 0100 1000 0001 1002 0103 0014 0001 1004 0103 0012 0001 0010 0100 1000 0001 1
  • 24. 분산시스템 연구실 6.3.3 pivoting example 1/2           =           ⋅           ⋅           − −⋅                     =           ⋅           ⋅           − −           =           ⋅                     =           220 040 888 888 442 484 001 010 100 102/1 014/1 001 010 100 001 040 220 888 888 442 484 001 010 100 102/1 014/1 001 484 442 888 888 442 484 001 010 100 888 442 484 888 442 484
  • 25. 분산시스템 연구실 6.3.3 pivoting example 2/2           ⋅           =           ⋅                     ⋅           ⋅           =           ⋅           ⋅                     =           ⋅           ⋅           ⋅           − −⋅           −           =           ⋅           ⋅           − −⋅           ⋅           − 200 040 888 12/12/1 014/1 001 888 442 484 010 001 100 200 040 888 12/10 010 001 102/1 014/1 001 888 442 484 001 010 100 010 100 001 200 040 888 888 442 484 001 010 100 010 100 001 102/1 014/1 001 12/10 010 001 200 040 888 888 442 484 001 010 100 104/1 012/1 001 010 100 001 12/10 010 001
  • 26. 분산시스템 연구실 6.3.4 pivoting procedure                                                                                                                                                 0010 0100 0001 1000 2220 3100 0040 8888 1008/1 0102/1 0014/1 0001 0001 0100 0010 1000 0040 3100 2220 8888 1002/1 0108/1 0014/1 0001 0001 0100 0010 1000 4484 4211 4442 8888 1000 0100 0010 0001 1000 0100 0010 0001 8888 4211 4442 4484 1000 0100 0010 0001                                                                                                                 0100 0010 0001 1000 2000 2200 0040 8888 12/102/1 012/18/1 0014/1 0001 0100 0010 0001 1000 3100 2200 0040 8888 1002/1 012/18/1 0014/1 0001 0010 0100 0001 1000 2200 3100 0040 8888 102/18/1 0102/1 0014/1 0001   
  • 27. 분산시스템 연구실 6.3.5 pivoting code Function [L, U, P] = LU_pivot(A) [n, n1] = size(A); L=eye(n); P=eye(n); U=A; For j = 1:n [pivot m] = max(abs(U(j:n, j))); m = m+j-1; if m ~= j % interchange rows m and j in U % interchange rows m and j in P if j >= 2 % interchange rows m and j in columns 1:j-1 of L end end for i = j+1:n L(i, j) = U(i, j) / U(j, j); U(i, :) = U(i, :) – L(i, j)*U(j, :); end end
  • 29. 분산시스템 연구실 6.4.1 Theory for Direct LU factorization )( )( 00 0 0 00 1111 1111 2211 21 22221 11211 222121111 222121222212211121 11112111111 21 22221 11211 222 11211 21 2221 11 jiululula jiululula ululule aaa aaa aaa eululul ululululul ululul aaa aaa aaa u uu uuu lll ll l jjijjjjijiij ijiijiiijiij nnnnnnnnnn nnnn n n nnnnn nn n nnnn n n nn n n nnnn ≥⋅+⋅++⋅= ≤⋅+⋅++⋅= ⋅++⋅+⋅=             =             ⋅+⋅⋅ ⋅+⋅⋅+⋅⋅ ⋅⋅⋅             =             •             −− −−                    
  • 30. 분산시스템 연구실 6.4.2 Type of Direct LU factorization • 조건의 수 = N^2 변수의 수 = N(N+1)/2 + N(N+1)/2 = N^2+N • Doolittle 방식 L 의 대각 원소가 1 • Crout 방식 U 의 대각 원소가 1 • Cholesky 방식 L 과 U 의 같은 위치 대각 원소끼리 같 음
  • 31. 분산시스템 연구실 6.4.3 Doolittle example           =           •                     =           •                     =           •                     =           •           64325 32204 541 300 1240 541 135 014 001 64325 32204 541 00 1240 541 135 014 001 64325 32204 541 00 0 541 15 014 001 64325 32204 541 00 0 1 01 001 33 33 2322 32 33 2322 131211 3231 21 u u uu l u uu uuu ll l
  • 32. 분산시스템 연구실 6.4.4 Doolittle code Function [L, U] = Doolittle(A) [n, m] = size(A); U = zeros(n, m); L = eye(n); for k = 1:n U(k, k) = A(k, k) – L(k, 1:k-1)*U(1:k-1, k); for j = k+1:n U(k, j) = A(k, j) – L(k, 1:k-1)*U(1:k-1, j); L(j,k)= (A(j,k)– L(j, 1:k-1)*U(1:k-1, k))/U(k,k); end end
  • 33. 분산시스템 연구실 6.4.5 Cholesky example           =           •                     =           •                     =           •                     =           •           64325 32204 541 300 620 541 365 024 001 64325 32204 541 00 620 541 65 024 001 64325 32204 541 00 0 541 5 04 001 64325 32204 541 00 00 00 3333 33 2322 3332 22 33 2322 131211 333231 2221 11 xx x ux xl x x ux uux xll xl x
  • 34. 분산시스템 연구실 6.4.6 Cholesky code Function [L, U] = Cholesky(A) % A is assumed to be symmetric. % L is computed, and U = L’ [n, m] = size(A); L = zeros(n, n); for k = 1:n L(k, k) = sqrt( A(k,k) – L(k, 1:k-1)*L(k, 1:k-1)’ ); for i = k+1:n L(i, k) = (A(i,k) – L(i, 1:k-1)*L(k, 1:k-1)’)/L(k, k); end end
  • 36. 분산시스템 연구실 Contents • 6.5 Applications of LU Factorization – Linear Equations – Tridiagonal System – Determinant of a Matrix – Inverse of a Matrix • 6.6 MATLAB’s Methods – lu, chol, det, inv
  • 37. 분산시스템 연구실 6.5.1 Solving Systems of Linear Equations • Ax = b → LUx = b → Ly = b – A = LU, Ux = y • Solve the system Ly = b for y – Forward substitution – y1 → y2 → …… yn • Solve the system Ux = y for x – Backward substitution – xn → xn-1 → …… x1 • If pivoting has been used – Ax=b → PAx=Pb, where PA=LU, Pb=c
  • 39. 분산시스템 연구실 Ex 6.8 Solving Electrical Circuit for Several Voltages 11 22 33 3 2 1 0.8044/2510(4/5)]-(56/25)(1/30)[-20 2.2456/255)40/3)(3/12(80 0.804/532/40 onsubstitutibackwardBy 32 80 0 4000 3/503/1250 1020-30 ix ix ix x x x ←=== ←==+= ←===           =                     − − ⇒= yUx
  • 40. 분산시스템 연구실 MATLAB Function to Solve the Linear System LUx=b function x = LU_Solve(L, U, b) % Function to solve the equation L U x = b % L --> Lower triangular matrix (with 1's on diagonal) % U --> Upper triangular matrix % b --> Right-hand side vector [n m] = size(L); z = zeros(n,1); x = zeros(n,1); % Solve L z = b using forward substitution z(1) = b(1); for i = 2:n z(i) = b(i) - L(i, 1:i-1) * z(1:i-1); end % Solve U x = z using back substitution x(n) = z(n) / U(n, n); for i = n-1 : -1 : 1 x(i) = (z(i) - U(i,i+1:n) * x(i+1:n)) / U(i, i); end
  • 41. 분산시스템 연구실 6.5.2 Solving a Tridiagonal System [ ] [ ] [ ] [ ] [ ] [ ]54-32-1 r)bb,dd,_solve(a,LU_tridiag 1111-143240 2173151 4324005214 89 6 5 23 7 214000 57300 02320 001154 00041 = = =−= ⇒= ==                 − − =                 −=⇒= x x ddb d bba rArAx Ex 6.9
  • 42. 분산시스템 연구실 MATLAB Function for Solving a Tridiagonal System function x = LU_tridiag_solve(a, d, b, r) % Function to solve the eqquation A x = r % LU factorization of A is expressed as % Lower bidiagonal matrix: % diagonal is 1s, lower diagonal is b; b(1) = 0. % Upper bidiagonal matrix: diagonal is d, upper diagonal is a % Right-hand-side vector is r % Solve L z = r using forward substitution n = length(d); z(1) = r(1); for i = 2:n z(i) = r(i) - b(i) * z(i-1); end % Solve U x = z using back substitution x(n) = z(n) / d(n); for i = n-1 : -1 : 1 x(i) = (z(i) - a(i) * x(i+1)) / d(i); end
  • 43. 분산시스템 연구실 6.5.3 Determinant of a Matrix ionfactorizatLUtheoccurredthat esinterchangrowofnumbertheiswhere )1()det( thatsoused,ispivotingIf )det(then,If 11 11 k ul ul n i ii n i ii k n i ii n i ii ∏∏ ∏∏ == == −= = == A LUPA ALUA -1 8)8)(1)(1()det( 800 510 211 111 012 001 121 112 211 332211 −=−==           − − =           −− −=⇒           − − − = uuuA ULA Ex 6.10
  • 44. 분산시스템 연구실 6.5.4 Inverse of a Matrix • The inverse of an n-by-n matrix A Axi=ei (i=1, … ,n) – ei=[0 0 … 1 … 0 0]’, where the 1 appears in the ith position – X whose columns are the solution vectors x1, … , xn is A-1
  • 45. 분산시스템 연구실 Ex 6.11 Finding a Matrix Inverse Using LU Factorization           =⇒           =                     −− −           =                     −− − =           =           −− −=           = 113 012 001 0 0 1 111 012 001 ofcolumningcorrespondon,substitutiforward:ofcolumnEach 100 010 001 111 012 001 forSolve 800 51-0 21-1 111 012 001 121- 112- 21-1 31 21 11 333231 232221 131211 Y IY YILY ULA y y y yyy yyy yyy
  • 46. 분산시스템 연구실 Ex 6.11 Finding a Matrix Inverse Using LU Factorization           −− − ==           =                     − − == − 8/18/18/3 8/58/38/1 8/38/58/1 ofcolumningcorrespondon,substitutiback:XofcolumnEach 113 012 001 800 510 211 then;forSolve 1 333231 232221 131211 1 AX Y XAXYUX xxx xxx xxx -
  • 47. 분산시스템 연구실 MATLAB Function function x = LU_Solve_Gen(L, U, B) % Function to solve the equation L U x = B % L --> Lower triangular matrix (1's on diagonal) % U --> Upper triangular matrix % B --> Right-hand-side matrix [n n2] = size(L); [m1 m] = size(B); % Solve L z = B using forward substitution for j = 1:m z(1, j) = b(1, j); for i = 2 : n z(i,j) = B(i,j) - L(i, 1:i-1) * z(1:i-1, j); end end % Solve U x = z using back substitution for j = 1:m x(n, j) = z(n, j) / U(n, n); for i = n-1 : -1 : 1 x(i,j) = (z(i,j)-U(i,i+1:n)*x(i+1:n,j)) / U(i,i); end end
  • 49. 분산시스템 연구실 6.6 MATLAB’s Method • 4 built-in functions – LU decomposition of a square matrix A • [L, U] = lu(A), [L, U, P] = lu(A) – Cholesky factorization • chol – Determinant of a matrix • det – Inverse of a matrix • inv