ALGEBRAIC EQUATIONS

xji = N − 1,N −2,...,1(2.3.7)44Chapter 2. Solution of Linear Algebraic EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.[r]

(called diagonal dominance)thenitcan be shown that the algorithmcannot encountera zero pivot.It is possible to construct special examples in which the lack of pivoting in thealgorithmcausesnumerical instability. In practice, however, such instability isalmostnever encountered — unlike the general ma[r]

m2=(m+1) >> 1;pp=g[m1];qq=h[m1];for (j=1;j<=m2;j++) {pt1=g[j];pt2=g[k];qt1=h[j];qt2=h[k];g[j]=pt1-pp*qt2;g[k]=pt2-pp*qt1;h[j]=qt1-qq*pt2;h[k ]=qt2-qq*pt1;}} Back for another recurrence.nrerror("toeplz - should not arrive here!");}If you arein the businessof solvingverylarge Toep[r]

for (sum=x[i],k=i+1;k<=n;k++) sum -= a[k][i]*x[k];98Chapter 2. Solution of Linear Algebraic EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by N[r]

b2···bM(2.0.3)By convention, the first index on an element aijdenotes its row, the secondindex its column. For most purposes you don’t need to know how a matrix is storedin a computer’s physical memory; you simply reference matrix elements by theirtwo-dimensional addresses, e.g., a34= a[3][4]. We[r]

• Interchanging any two columns of A gives the same solution set onlyif we simultaneously interchange corresponding rows of the x’s and ofY. In other words, this interchange scrambles the order of the rows inthe solution. If we do this, we will need to unscramble the solution byrestoring the rows to[r]

98Chapter 2. Solution of Linear Algebraic EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for inte[r]

58Chapter 2. Solution of Linear Algebraic EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for inte[r]

1b2b3b4(2.2.1)Here the primes signify that the a’s and b’s do not have their original numericalvalues, but have been modified by all the row operations in the elimination to thispoint. The procedure up to this point is termed Gaussian elimination.42Chapter 2. Solution of Linear Algebraic