COMPUTER OPTIMIZATION

Transcription

1 COMPUTER OPTIMIZATION Storage Optimization: Since Normal Matrix is a symmetric matrix, store only half of it in a vector matrix and develop a indexing scheme to map the upper or lower half to the vector. Vector Storage PLATE 22-1

2 INDEXING SCHEMES Upper triangular matrix to vector: Index( i, j ) = 1/2 [ j ( j-1 )] + i Example of indexing (2, 3) element: Index( 2,3 ) = 1/2 {3 (3-1)}+2 = 5 Lower triangular matrix to vector: Index( i, j ) = 1/2 [ i ( i-1 )] + j Example of indexing (3, 2) element: Index( 3,2 ) = 1/2 {3 (3-1)}+2 = 5 PLATE 22-2

3 INDEXING SCHEMES Use a MAPPING TABLE to reduce computation of indices. Upper Triangular Storage VI is a mapping table indicating the storage location immediately prior to the first element of each column. Lower Triangular Storage VI is a mapping table indicating the storage location immediately prior to the first element of each row. Computing a Mapping Table VI(1) = 0 for i going from 2 to the number of unknowns VI(i) = VI(i-1) + i-1 PLATE 22-3

4 INDEXING USING A MAPPING TABLE Upper Triangular Matrix Index( i, j ) = VI(j) + I Example of indexing (2, 3) element: Index( 2,3 ) = VI(3) + 2 = = 5 Lower Triangular Matrix Index( i, j ) = VI(j) + I Example of indexing (3, 2) element: Index( 3,2 ) = VI(3) + 2 = = 5 PLATE 22-4

5 TIME COMPARISONS Method Time Extra Method A: Using Function A (sec) storage i from 1 to j from i to 1000 A k = Index( i,j ) B bytes per element C bytes per element D full matrix required Method B: Using Function B i from 1 to 1000 j from i to 1000 k = Index( i,j ) Method C: Direct access using mapping table: i from 1 to 1000 j from i to 1000 k = Vi(j) + I Function A: Index = j*(j-1)/2 + i Function B: Index = Vi(j) + i; Method D: Direct matrix element access. i from 1 to 1000 j from 1 to 1000 k = A[i,j] Note: Method D uses direct access to a full matrix and thus is the fastest at access to element, but most costly in storage requirements. Methods A, B and C only require storage of upper triangle of normal matrix, and that use of a mapping table almost as fast as direct access. PLATE 22-5

6 PLATE 22-6

7 DIRECT FORMATION OF UPPER-TRIANGULAR NORMAL MATRIX Adding a single row of an a matrix directly to normal matrix and constants matrix. ADVANTAGES: T Avoids formation of intermediate matrices A, A, T W, L, and A W. Faster times in development of normal equations. Steps: 1. Zero the normal and constants matrix. 2. Zero a single row of the coefficient matrix. 3. Based on the values of a single row of the coefficient matrix, add the proper value to the appropriate normal and constant elements. 4. Repeat Steps 3 and 4 for all observations. PLATE 22-7

8 COMPUTER CODE FOR MATRIX FORMATION (BASIC) VI is a mapping table, T N is the normal matrix (A WA), T C is the constants matrix (A WL) W is the weight matrix unknown is the number of unknown parameters. For i = 1 to unknown ix = VI(i): ixi = ix + i N(ixi) = N(ixi) + A(i)^2 * W(i) C(i) = C(i) + A(i)*W(i)*L(i) For j = i+1 to unknown N(ix+j) = N(ix+j) + A(i)*W(i)*A(j) Next j Next i PLATE 22-8

9 COMPUTER CODE FOR MATRIX FORMATION (C) vi is a mapping table, T n is the normal matrix (A WA), T c is the constants matrix (A WL) w is the weight matrix unknown is the number of unknown parameters. for ( i=1; i<=unknown; i++) { ix = vi[i]; ixi = ix + i; n[ixi] += a[i] * a[i] * w[i]; c[i] += a[i] * w[i] * l[i]; for ( j=i+1; j<=unknown; j++) n[ix+j] += a[i] * w[i] * a[j]; }// for i PLATE 22-9

10 COMPUTER CODE FOR MATRIX FORMATION (FORTRAN) VI is a mapping table, T N is the normal matrix (A WA), T C is the constants matrix (A WL) W is the weight matrix unknown is the number of unknown parameters. Do 100 i = 1, unknown ix = VI(i) ixi = ix + i N(ixi) = N(ixi) + A(i)**2 * W(i) C(i) = C(i) + A(i) * W(i) * L(i) Do 100 j = i+1, unknown N(ix+j) = N(ix+j) + A(i) * W(i) * A(j) 100 Continue PLATE 22-10

11 COMPUTER CODE FOR MATRIX FORMATION (PASCAL) VI is a mapping table, T N is the normal matrix (A WA), T C is the constants matrix (A WL) W is the weight matrix, and unknown is the number of unknown parameters. For i := 1 to unknown do begin ix := VI[i]; ixi := ix + i; N[ixi] := N[ixi] + Sqr(A[i]) * W[i]; C[i] := C[i] + A[i] * W[i] * L[i]; For j := i+1 to unknown do N[ix+j] := N[ix+j] + A[i] * W[i] * A[j] End; { for i } PLATE 22-11

12 CHOLESKY DECOMPOSITION Method of solving a system of equations that decreases computations over full inversing of matrix. Since the normal matrix is a Hermitian matrix, it can be written as the product of lower, L, and upper, U, triangular T matrices where the L = U, or: l ii l ii i 1 k 1 l 2 ik T T N = LU = LL = U U Also the resultant decomposed matrix can overwrite original normal matrix stored as a vector. Pseudo Code for i = 1, 2,, number of unknowns 1 2 For j = i+1, i+2,, number of unknowns i 1 l ji l ji k 1 l ik l jk. l ii PLATE 22-12

13 CHOLESKY DECOMPOSITION (BASIC) s is summing variable FOR i = 1 TO unknown ix = VI(i): ixi = ix + i: s = 0# FOR k = 1 TO i - 1 s = s + N(ix + k) ^ 2 NEXT k N(ixi) = SQR(N(ixi) - s) FOR j = i + 1 TO unknown s = 0#: jx = vi(j) FOR k = 1 TO i - 1 s = s + N(jx + k) * N(ix + k) NEXT k N(jx + i) = (N(jx + i) - s) / N(ixi) NEXT j NEXT I PLATE 22-13

14 CHOLESKY DECOMPOSITION (C) s is summing variable for ( i=1; i<=unknown; i++ ) { ix = vi[i]; ixi = ix+i; s = 0.0; for ( k=1; k<i; k++ ) s += n[ix+k]*n[ix+k]; n[ixi] = sqrt( n[ixi] - s ); for ( j=i+1; j<=unknown; j++ ) { s = 0; jx = vi[j]; for ( k=1; k<i; k++) s += n[jx+k] * n[ix+k]; n[jx+i] = ( n[jx+i] - s ) / n[ixi]; } // for j } // for i PLATE 22-14

15 CHOLESKY DECOMPOSITION (FORTRAN) s is summing variable Do 30 i = 1,Unknown ix = VI(i) i1 = i-1 S = 0.0 Do 10 k=1, i1 10 s = s + N( ix+k )**2 N(ix+i) = Sqrt( N(ix+i)-s ) Do 30 j = i+1, Unknown s = 0.0 Do 20 k = 1, i1 20 s = s + N( VI[j]+k ) * N( ix+k ) N(VI[j]+i) = (N(VI[j]+i)-s) / N(ix+i) 30 Continue PLATE 22-15

16 CHOLESKY DECOMPOSITION (PASCAL) s is summing variable For i := 1 to unknown do Begin ix := VI[i]; ixi := ix+i; S := 0; For k := 1 to Pred( i ) do S := S + Sqr( N[ix+k] ); N[ixi] := Sqrt( N[ixi] -S ); For j := Succ(i) to unknown do Begin S := 0; jx := Vi[j]; For k := 1 to Pred(i) do S := S + N[jx+k]*N[ix+k]; N[jx+i] := ( N[jx+i] - S ) / N[ixi] End; { for j} End; { for i } PLATE 22-16

17 SOLUTION OF A TRIANGULAR MATRIX SYSTEM l l 11 l 21 l 31 l n1 x 1 c 1 l 21 l l 22 l 32 l n2 x 2 c 2 LUX l 31 l 32 l l 33 l n3 x 3 c 3 C 0 l n1 l n2 l n3 l nn l nn x n c n Equation can be written as: and LY = C UX = Y Solve for Y using forward-substitution. Then solve for X using backward-substitution. PLATE 22-17

18 Pseudo Code: FORWARD SUBSTITUTION 1. Solve for y as: 1 y 1 c 1 l Substitute this value into row 2, and compute y as: 2 y 2 c 2 l 21 y 1 l Repeat this procedure until all values for y are found using the algorithm: i 1 y i c i k 1 l ik y k l ii PLATE 22-18

19 Pseudo Code: BACKWARD SUBSTITUTION 1. Compute x as: n x n y n l nn 2. Solve for x as: n-1 x n 1 y n 1 l nn x n l n 1 n 1 3. Repeat this procedure until all unknowns are computed using the algorithm: n x k y k j k 1 l kj y j l kk PLATE 22-19

20 SUBSTITUTION PROCESS (BASIC) Rem Forward Substitution For i = 1 to Unknown ix = VI(i) C(i) = C(i) / N(ix+i) For j = i+1 to Unknown C(j) = C(j) - N(ix+j) * C(i) Next j Next i Rem Backward Substitution For i = Unknown to 1 Step -1 ix = VI(i) For j = Unknown To i+1 Step -1 C(i) = N(ix+j) * C(j) Next j C(i) = C(i) / N(ix+i) Next i PLATE 22-20

21 SUBSTITUTION PROCESS (C) //Forward Substitution for (i=1; i<=unknown; I++) { ix = vi[i]; c[i] = c[i]/n[ix+i]; for (j=i+1; j<=unknown; j++) { c[j] -= n[ix+j]*c[i]; } //for j } //for i //backward substitution for (i=unknown; i>=1; i--) { ix = vi[i]; for (j=unknown; j>=i+1; j--) { c[i] -= n[ix+j] * c[j]; } //for j c[i] = c[i]/n[ix+i]; } // for i PLATE 22-21

22 SUBSTITUTION PROCESS (FORTRAN) C Forward Substitution Do 100 i = 1,Unknown ix = VI(i) C(i) = C(i) / N(ix+i) Do 100 j = i+1, Unknown C(j) = C(j) - N(ix+j) * C(i) 100 Continue C Backward Substitution Do 110 i = Unknown, 1,-1 ix = VI(i) Do 120 j = Unknown, i+1, C(i) = C(i) - N(Ix+j) * C(j) C(i) = C(i) / N(Ix+i) 110 Continue PLATE 22-22

23 SUBSTITUTION PROCESS (PASCAL) {Forward Substitution} For i := 1 to Unknown Do Begin ix := VI[i]; C[i] := C[i] / N[ix+i]; For j := i+1 to Unknown Do C[j] := C[j] - N[ix+j] * C[i]; End; { for i } {Backward Substitution} For i := Unknown DownTo 1 do Begin ix := VI[i]; For j := Unknown DownTo i+1 Do C[i] := C[i] - N[ix+j] * C[j]; C[i] := C[i] / N[ix+i]; End; { for i } PLATE 22-23

24 COMPUTING THE INVERSE MATRIX FROM A CHOLESKY FACTOR Inverse matrix necessary to compute post-adjustment statistics. Inverse of full matrix is found by: 1. Computing inverse of Cholesky factor calling S 2. The inverse of the full matrix = S S T Pseudo code for computing inverse of Cholesky factor. for i going from the number of unknowns down to 1 for k going from number of unknowns down to i+1 for j going from i+1 to k sum the product N(i, j) * N(j, k) N(i, k) = -S / N(i, I) N(i, i) = 1/N(i, i) PLATE 22-24

25 CODE FOR COMPUTING INVERSE (BASIC) {Inverse} For i = Unknown To 1 Step -1 ix = VI(i): ixi = ix + i For k = Unknown To i+1 Step -1 S = 0! For j = i+1 To k S = S + N(ix+j) * N(VI(j)+k) Next j N(ix+k) = -S / N(ixi) Next k N(ixi) = 1.0 / N(ixi) Next i {Inverse * Transpose of Inverse} For j = 1 to Unknown ixj = VI(j) For k = j to Unknown S = 0! For i = k to Unknown S = S + N(VI(k)+i) * N(ixj+i) Next i N(ixj+k) = S Next k Next j PLATE 22-25

26 CODE FOR COMPUTING INVERSE (C) {Inverse} for ( i=unknown; i>=1; i--) { ix = vi[i]; ixi = ix + i; for( k=unknown; k>=i+1; k--) { s = 0.0; for ( j=i+1; j<=k; j++) s += n[ix+j] * n[vi[j]+k]; n[ix+k] = -s / n[ixi]; } // for k n[ixi] = 1.0 / n[ixi]; } // for i {inverse * transpose of inverse} for (j=1; j<=unknown; j++) { ixj = vi[j]; for (k=j; k<=unknown; k++) { s = 0.0; For (i=k; i<=unknown; i++) s += n[vi[k]+i] * n[ixj+i]; n[ixj+k] = s } // for k } //for j PLATE 22-26

27 CODE FOR COMPUTING INVERSE (FORTRAN) C Inverse Do 10 i = Unknown, 1,-1 ix = VI(i) ixi = ix + i Do 11 k = Unknown, i+1,-1 S = 0.0 Do 12 j = i+1,k 12 S = S + N(ix+j) * N(Vi(j)+k) N(ix+k) = -S / N(ixi) 11 Continue N(ixi) = 1.0 / N(ixi) 10 Continue C Inverse * Transpose of Inverse} Do 20 j = 1, Unknown ixj = VI(j) Do 21 k = j, Unknown S = 0.0 Do 22 i = k,unknown 22 S = S + N(Vi(k)+i) * N(ixj+i) N(ixj+k) = S 21 Continue 20 Continue PLATE 22-27

28 CODE FOR COMPUTING INVERSE (PASCAL) {Inverse} For i := Unknown DownTo 1 do Begin ix := VI[i]; ixi := ix + i; For k := Unknown DownTo i+1 Do Begin S := 0.0; For j := i+1 To k Do S := S + N[ix+j] * N[Vi[j]+k]; N[ix+k] := -S / N[ixi]; End; { For k } N[ixi] := 1.0 / N[ixi]; End; { For i } {Inverse * Transpose of Inverse} For j := 1 to Unknown Do Begin ixj := VI[j]; For k := j to Unknown Do Begin S := 0.0; For i := k to Unknown Do S := S + N[Vi[k]+i] * N[ixj+i]; N[ixj+k] := S End; { For k } End; { For j } PLATE 22-28

29 SPARENESS AND OPTIMIZATION EXAMPLE NETWORK: All lines have measured lengths. Angles are measured as shown. Non-zero element in normal matrix (Note: x and y elements are combined.) PLATE 22-29

30 SAME NORMAL MATRIX WITH STATIONS IN DIFFERENT ORDER Note that known off-diagonal zero elements are to the left of non-zero elements. PLATE 22-30

31 OPTIMIZING CHOLESKY DECOMPOSITION TO TAKE ADVANTAGE OF REORDERING Method by which Cholesky accesses matrix: not accessed computed and accessed currently accessed yet to be accessed Full Operations 1) s = n 51 + n n54 s = n 53 + n54 Optimized Operations ) n = square root of ( n - s ) n = square root of (n - s) ) j = 6 j = 6 a) s = n *n + n *n + + n *n no operations necessary b) n = (n - s)/n j = 7 j = 7 a) s = n *n + n *n + + n *n no operations necessary b) n = (n - s)/n j = 8 j = 8 a) s = n *n + n *n + + n *n s = n *n + n *n b) n = (n - s)/n n = (n - s)/n PLATE 22-31

32 REORDERING Sta Connectivity Deg ,3, ,3, ,2,4,7, ,5,6,7, ,6, ,5, ,4,5,6, ,2,3,4, Deg indicates the number stations connected by observations to the Sta. Steps: 1. Chose station with lowest degree. 2. Remove station from connectivity list and adjust degrees for each station. 3. Repeat 1 and 2 until all stations are used. PLATE 22-32