# include # include # include # include # include "r83s.h" /******************************************************************************/ int i4_log_10 ( int i ) /******************************************************************************/ /* Purpose: i4_log_10() returns the integer part of the logarithm base 10 of an I4. Example: I I4_LOG_10 ----- -------- 0 0 1 0 2 0 9 0 10 1 11 1 99 1 100 2 101 2 999 2 1000 3 1001 3 9999 3 10000 4 Discussion: I4_LOG_10 ( I ) + 1 is the number of decimal digits in I. Licensing: This code is distributed under the MIT license. Modified: 23 October 2007 Author: John Burkardt Parameters: Input, int I, the number whose logarithm base 10 is desired. Output, int I4_LOG_10, the integer part of the logarithm base 10 of the absolute value of X. */ { int i_abs; int ten_pow; int value; if ( i == 0 ) { value = 0; } else { value = 0; ten_pow = 10; i_abs = abs ( i ); while ( ten_pow <= i_abs ) { value = value + 1; ten_pow = ten_pow * 10; } } return value; } /******************************************************************************/ int i4_max ( int i1, int i2 ) /******************************************************************************/ /* Purpose: I4_MAX returns the maximum of two I4's. Licensing: This code is distributed under the MIT license. Modified: 29 August 2006 Author: John Burkardt Parameters: Input, int I1, I2, are two integers to be compared. Output, int I4_MAX, the larger of I1 and I2. */ { int value; if ( i2 < i1 ) { value = i1; } else { value = i2; } return value; } /******************************************************************************/ int i4_min ( int i1, int i2 ) /******************************************************************************/ /* Purpose: I4_MIN returns the smaller of two I4's. Licensing: This code is distributed under the MIT license. Modified: 29 August 2006 Author: John Burkardt Parameters: Input, int I1, I2, two integers to be compared. Output, int I4_MIN, the smaller of I1 and I2. */ { int value; if ( i1 < i2 ) { value = i1; } else { value = i2; } return value; } /******************************************************************************/ double r8_uniform_01 ( int *seed ) /******************************************************************************/ /* Purpose: R8_UNIFORM_01 returns a unit pseudorandom R8. Discussion: This routine implements the recursion seed = 16807 * seed mod ( 2^31 - 1 ) r8_uniform_01 = seed / ( 2^31 - 1 ) The integer arithmetic never requires more than 32 bits, including a sign bit. If the initial seed is 12345, then the first three computations are Input Output R8_UNIFORM_01 SEED SEED 12345 207482415 0.096616 207482415 1790989824 0.833995 1790989824 2035175616 0.947702 Licensing: This code is distributed under the MIT license. Modified: 11 August 2004 Author: John Burkardt Reference: Paul Bratley, Bennett Fox, Linus Schrage, A Guide to Simulation, Springer Verlag, pages 201-202, 1983. Pierre L'Ecuyer, Random Number Generation, in Handbook of Simulation edited by Jerry Banks, Wiley Interscience, page 95, 1998. Bennett Fox, Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators, ACM Transactions on Mathematical Software, Volume 12, Number 4, pages 362-376, 1986. P A Lewis, A S Goodman, J M Miller, A Pseudo-Random Number Generator for the System/360, IBM Systems Journal, Volume 8, pages 136-143, 1969. Parameters: Input/output, int *SEED, the "seed" value. Normally, this value should not be 0. On output, SEED has been updated. Output, double R8_UNIFORM_01, a new pseudorandom variate, strictly between 0 and 1. */ { int k; double r; k = *seed / 127773; *seed = 16807 * ( *seed - k * 127773 ) - k * 2836; if ( *seed < 0 ) { *seed = *seed + 2147483647; } r = ( ( double ) ( *seed ) ) * 4.656612875E-10; return r; } /******************************************************************************/ void r83s_cg ( int n, double a[], double b[], double x[] ) /******************************************************************************/ /* Purpose: R83S_CG uses the conjugate gradient method on an R83S system. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. The matrix A must be a positive definite symmetric band matrix. The method is designed to reach the solution after N computational steps. However, roundoff may introduce unacceptably large errors for some problems. In such a case, calling the routine again, using the computed solution as the new starting estimate, should improve the results. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 09 July 2014 Author: John Burkardt Reference: Frank Beckman, The Solution of Linear Equations by the Conjugate Gradient Method, in Mathematical Methods for Digital Computers, edited by John Ralston, Herbert Wilf, Wiley, 1967, ISBN: 0471706892, LC: QA76.5.R3. Parameters: Input, int N, the order of the matrix. N must be positive. Input, double A[3], the matrix. Input, double B[N], the right hand side vector. Input/output, double X[N]. On input, an estimate for the solution, which may be 0. On output, the approximate solution vector. */ { double alpha; double *ap; double beta; int i; int it; double *p; double pap; double pr; double *r; double rap; /* Initialize AP = A * x, R = b - A * x, P = b - A * x. */ ap = r83s_mv ( n, n, a, x ); r = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { r[i] = b[i] - ap[i]; } p = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { p[i] = b[i] - ap[i]; } /* Do the N steps of the conjugate gradient method. */ for ( it = 1; it <= n; it++ ) { /* Compute the matrix*vector product AP=A*P. */ free ( ap ); ap = r83s_mv ( n, n, a, p ); /* Compute the dot products PAP = P*AP, PR = P*R Set ALPHA = PR / PAP. */ pap = r8vec_dot_product ( n, p, ap ); pr = r8vec_dot_product ( n, p, r ); if ( pap == 0.0 ) { free ( ap ); break; } alpha = pr / pap; /* Set X = X + ALPHA * P R = R - ALPHA * AP. */ for ( i = 0; i < n; i++ ) { x[i] = x[i] + alpha * p[i]; } for ( i = 0; i < n; i++ ) { r[i] = r[i] - alpha * ap[i]; } /* Compute the vector dot product RAP = R*AP Set BETA = - RAP / PAP. */ rap = r8vec_dot_product ( n, r, ap ); beta = - rap / pap; /* Update the perturbation vector P = R + BETA * P. */ for ( i = 0; i < n; i++ ) { p[i] = r[i] + beta * p[i]; } } /* Free memory. */ free ( p ); free ( r ); return; } /******************************************************************************/ double *r83s_dif2 ( int m, int n ) /******************************************************************************/ /* Purpose: R83S_DIF2 returns the DIF2 matrix in R83S format. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Properties: A is banded, with bandwidth 3. A is tridiagonal. Because A is tridiagonal, it has property A (bipartite). A is a special case of the TRIS or tridiagonal scalar matrix. A is integral, therefore det ( A ) is integral, and det ( A ) * inverse ( A ) is integral. A is Toeplitz: constant along diagonals. A is symmetric: A' = A. Because A is symmetric, it is normal. Because A is normal, it is diagonalizable. A is persymmetric: A(I,J) = A(N+1-J,N+1-I). A is positive definite. A is an M matrix. A is weakly diagonally dominant, but not strictly diagonally dominant. A has an LU factorization A = L * U, without pivoting. The matrix L is lower bidiagonal with subdiagonal elements: L(I+1,I) = -I/(I+1) The matrix U is upper bidiagonal, with diagonal elements U(I,I) = (I+1)/I and superdiagonal elements which are all -1. A has a Cholesky factorization A = L * L', with L lower bidiagonal. L(I,I) = sqrt ( (I+1) / I ) L(I,I-1) = -sqrt ( (I-1) / I ) The eigenvalues are LAMBDA(I) = 2 + 2 * COS(I*PI/(N+1)) = 4 SIN^2(I*PI/(2*N+2)) The corresponding eigenvector X(I) has entries X(I)(J) = sqrt(2/(N+1)) * sin ( I*J*PI/(N+1) ). Simple linear systems: x = (1,1,1,...,1,1), A*x=(1,0,0,...,0,1) x = (1,2,3,...,n-1,n), A*x=(0,0,0,...,0,n+1) det ( A ) = N + 1. The value of the determinant can be seen by induction, and expanding the determinant across the first row: det ( A(N) ) = 2 * det ( A(N-1) ) - (-1) * (-1) * det ( A(N-2) ) = 2 * N - (N-1) = N + 1 Licensing: This code is distributed under the MIT license. Modified: 09 July 2014 Author: John Burkardt Reference: Robert Gregory, David Karney, A Collection of Matrices for Testing Computational Algorithms, Wiley, 1969, ISBN: 0882756494, LC: QA263.68 Morris Newman, John Todd, Example A8, The evaluation of matrix inversion programs, Journal of the Society for Industrial and Applied Mathematics, Volume 6, Number 4, pages 466-476, 1958. John Todd, Basic Numerical Mathematics, Volume 2: Numerical Algebra, Birkhauser, 1980, ISBN: 0817608117, LC: QA297.T58. Joan Westlake, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, John Wiley, 1968, ISBN13: 978-0471936756, LC: QA263.W47. Parameters: Input, int M, N, the order of the matrix. Output, double A[3], the matrix. */ { double *a; a = ( double * ) malloc ( 3 * sizeof ( double ) ); a[0] = -1.0; a[1] = +2.0; a[2] = -1.0; return a; } /******************************************************************************/ void r83s_gs_sl ( int n, double a[], double b[], double x[], int it_max ) /******************************************************************************/ /* Purpose: R83S_GS_SL solves an R83S system using Gauss-Seidel iteration. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. This routine simply applies a given number of steps of the iteration to an input approximate solution. On first call, you can simply pass in the zero vector as an approximate solution. If the returned value is not acceptable, you may call again, using it as the starting point for additional iterations. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int N, the order of the matrix. Input, double A[3], the R83S matrix. Input, double B[N], the right hand side of the linear system. Input/output, double X[N], an approximate solution to the system. Input, int IT_MAX, the maximum number of iterations to take. */ { int i; int it_num; /* No diagonal matrix entry can be zero. */ if ( a[1] == 0.0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83S_GS_SL - Fatal error!\n" ); fprintf ( stderr, " Zero diagonal entry\n" ); return; } for ( it_num = 1; it_num <= it_max; it_num++ ) { x[0] = ( b[0] - a[2] * x[1] ) / a[1]; for ( i = 1; i < n-1; i++ ) { x[i] = ( b[i] - a[0] * x[i-1] - a[2] * x[i+1] ) / a[1]; } x[n-1] = ( b[n-1] - a[0] * x[n-2] ) / a[1]; } return; } /******************************************************************************/ double *r83s_indicator ( int m, int n ) /******************************************************************************/ /* Purpose: R83S_INDICATOR sets up an R83s indicator matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 05 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Output, double R83_INDICATOR[3], the R83S indicator matrix. */ { double *a; a = ( double * ) malloc ( 3 * sizeof ( double ) ); a[0] = 3.0; a[1] = 2.0; a[2] = 1.0; return a; } /******************************************************************************/ void r83s_jac_sl ( int n, double a[], double b[], double x[], int it_max ) /******************************************************************************/ /* Purpose: R83S_JAC_SL solves an R83S system using Jacobi iteration. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. This routine simply applies a given number of steps of the iteration to an input approximate solution. On first call, you can simply pass in the zero vector as an approximate solution. If the returned value is not acceptable, you may call again, using it as the starting point for additional iterations. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int N, the order of the matrix. Input, double A[3], the R83S matrix. Input, double B[N], the right hand side of the linear system. Input/output, double X[N], an approximate solution to the system. Input, int IT_MAX, the maximum number of iterations to take. */ { int i; int it_num; double *xnew; xnew = ( double * ) malloc ( n * sizeof ( double ) ); /* No diagonal matrix entry can be zero. */ if ( a[1] == 0.0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83S_JAC_SL - Fatal error!\n" ); fprintf ( stderr, " Zero diagonal entry\n" ); return; } for ( it_num = 1; it_num <= it_max; it_num++ ) { /* Solve A*x=b: */ xnew[0] = b[0] - a[2] * x[1]; for ( i = 1; i < n-1; i++ ) { xnew[i] = b[i] - a[0] * x[i-1] - a[2] * x[i+1]; } xnew[n-1] = b[n-1] - a[0] * x[n-2]; /* Divide by the diagonal term, and overwrite X. */ for ( i = 0; i < n; i++ ) { xnew[i] = xnew[i] / a[1]; } for ( i = 0; i < n; i++ ) { x[i] = xnew[i]; } } free ( xnew ); return; } /******************************************************************************/ double *r83s_mtv ( int m, int n, double a[], double x[] ) /******************************************************************************/ /* Purpose: R83S_MTV multiplies a vector times an R83S matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 05 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the linear system. Input, double A[3], the R83S matrix. Input, double X[M], the vector to be multiplied. Output, double R83S_MTV[N], the product A'*x. */ { double *b; int i; int i_hi; int i_lo; int j; b = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { b[i] = 0.0; } for ( j = 0; j < n; j++ ) { i_lo = i4_max ( 0, j - 1 ); i_hi = i4_min ( m - 1, j + 1 ); for ( i = i_lo; i <= i_hi; i++ ) { b[j] = b[j] + x[i] * a[i-j+1]; } } return b; } /******************************************************************************/ double *r83s_mv ( int m, int n, double a[], double x[] ) /******************************************************************************/ /* Purpose: R83S_MV multiplies an R83S matrix times a vector. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int M, N, the number of rows and columns. Input, double A[3], the R83S matrix. Input, double X[N], the vector to be multiplied by A. Output, double R83S_MV[M], the product A * x. */ { double *b; int i; int i_hi; int i_lo; int j; b = ( double * ) malloc ( m * sizeof ( double ) ); for ( i = 0; i < m; i++ ) { b[i] = 0.0; } for ( j = 0; j < n; j++ ) { i_lo = i4_max ( 0, j - 1 ); i_hi = i4_min ( m - 1, j + 1 ); for ( i = i_lo; i <= i_hi; i++ ) { b[i] = b[i] + a[i-j+1] * x[j]; } } return b; } /******************************************************************************/ void r83s_print ( int m, int n, double a[], char *title ) /******************************************************************************/ /* Purpose: R83S_PRINT prints an R83S matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3], the R83S matrix. Input, char *TITLE, a title. */ { r83s_print_some ( m, n, a, 1, 1, m, n, title ); return; } /******************************************************************************/ void r83s_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi, int jhi, char *title ) /******************************************************************************/ /* Purpose: R83S_PRINT_SOME prints some of an R83S matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3], the R83S matrix. Input, int ILO, JLO, IHI, JHI, designate the first row and column, and the last row and column, to be printed. Input, char *TITLE, a title. */ { # define INCX 5 int i; int i2hi; int i2lo; int inc; int j; int j2; int j2hi; int j2lo; printf ( "\n" ); printf ( "%s\n", title ); /* Print the columns of the matrix, in strips of 5. */ for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX ) { j2hi = j2lo + INCX - 1; j2hi = i4_min ( j2hi, n ); j2hi = i4_min ( j2hi, jhi ); inc = j2hi + 1 - j2lo; printf ( "\n" ); printf ( " Col: " ); for ( j = j2lo; j <= j2hi; j++ ) { j2 = j + 1 - j2lo; printf ( "%7d ", j ); } printf ( "\n" ); printf ( " Row\n" ); printf ( " ---\n" ); /* Determine the range of the rows in this strip. */ i2lo = i4_max ( ilo, 1 ); i2lo = i4_max ( i2lo, j2lo - 1 ); i2hi = i4_min ( ihi, m ); i2hi = i4_min ( i2hi, j2hi + 1 ); for ( i = i2lo; i <= i2hi; i++ ) { /* Print out (up to) 5 entries in row I, that lie in the current strip. */ printf ( "%6d ", i ); for ( j2 = 1; j2 <= inc; j2++ ) { j = j2lo - 1 + j2; if ( i - j + 1 < 0 || 2 < i - j + 1 ) { printf ( " " ); } else { printf ( "%12f ", a[i-j+1] ); } } printf ( "\n" ); } } return; # undef INCX } /******************************************************************************/ double *r83s_random ( int m, int n, int *seed ) /******************************************************************************/ /* Purpose: R83S_RANDOM randomizes an R83S matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the linear system. Input/output, int *SEED, a seed for the random number generator. Output, double R83S_RANDOM[3], the R83S matrix. */ { double *a; a = r8vec_uniform_01_new ( 3, seed ); return a; } /******************************************************************************/ double *r83s_res ( int m, int n, double a[], double x[], double b[] ) /******************************************************************************/ /* Purpose: R83S_RES computes the residual R = B-A*X for R83S matrices. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3], the matrix. Input, double X[N], the vector to be multiplied by A. Input, double B[M], the desired result A * x. Output, double R83S_RES[M], the residual R = B - A * X. */ { int i; double *r; r = r83s_mv ( m, n, a, x ); for ( i = 0; i < m; i++ ) { r[i] = b[i] - r[i]; } return r; } /******************************************************************************/ double *r83s_to_r8ge ( int m, int n, double a[] ) /******************************************************************************/ /* Purpose: R83S_TO_R8GE copies an R83S matrix to an R8GE matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 04 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3], the R83S matrix. Output, double R83S_TO_R8GE[M*N], the R8GE matrix. */ { double *b; int i; int i_hi; int i_lo; int j; b = ( double * ) malloc ( m * n * sizeof ( double ) ); for ( j = 0; j < n; j++ ) { for ( i = 0; i < m; i++ ) { b[i+j*m] = 0.0; } } for ( j = 0; j < n; j++ ) { i_lo = i4_max ( 0, j - 1 ); i_hi = i4_min ( m - 1, j + 1 ); for ( i = i_lo; i <= i_hi; i++ ) { b[i+j*m] = a[i-j+1]; } } return b; } /******************************************************************************/ double *r83s_zeros ( int m, int n ) /******************************************************************************/ /* Purpose: R83S_ZEROS zeros an R83S matrix. Discussion: The R83S storage format is used for a tridiagonal scalar matrix. The vector A(3) contains the subdiagonal, diagonal, and superdiagonal values that occur on every row. RGE A(I,J) = R83S A[I-J+1]. Example: Here is how an R83S matrix of order 5, stored as (A1,A2,A3), would be interpreted: A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 A1 0 0 0 A3 A2 Licensing: This code is distributed under the MIT license. Modified: 18 August 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the linear system. Input/output, int *SEED, a seed for the random number generator. Output, double R83S_ZERO[3], the R83S matrix. */ { double *a; int i; a = ( double * ) malloc ( 3 * sizeof ( double ) ); for ( i = 0; i < 3; i++ ) { a[i] = 0.0; } return a; } /******************************************************************************/ double *r8ge_mtv ( int m, int n, double a[], double x[] ) /******************************************************************************/ /* Purpose: R8GE_MTV multiplies a vector times an R8GE matrix. Discussion: The R8GE storage format is used for a "general" M by N matrix. A physical storage space is made for each logical entry. The two dimensional logical array is mapped to a vector, in which storage is by columns. Licensing: This code is distributed under the MIT license. Modified: 06 March 2012 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, double A[M*N], the R8GE matrix. Input, double X[M], the vector to be multiplied by A. Output, double R8GE_MTV[N], the product A' * x. */ { double *b; int i; int j; b = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { b[i] = 0.0; for ( j = 0; j < m; j++ ) { b[i] = b[i] + a[j+i*m] * x[j]; } } return b; } /******************************************************************************/ double *r8ge_mv ( int m, int n, double a[], double x[] ) /******************************************************************************/ /* Purpose: R8GE_MV multiplies an R8GE matrix times a vector. Discussion: The R8GE storage format is used for a "general" M by N matrix. A physical storage space is made for each logical entry. The two dimensional logical array is mapped to a vector, in which storage is by columns. Licensing: This code is distributed under the MIT license. Modified: 28 February 2012 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, double A[M*N], the R8GE matrix. Input, double X[N], the vector to be multiplied by A. Output, double R8GE_MV[M], the product A * x. */ { double *b; int i; int j; b = ( double * ) malloc ( m * sizeof ( double ) ); for ( i = 0; i < m; i++ ) { b[i] = 0.0; for ( j = 0; j < n; j++ ) { b[i] = b[i] + a[i+j*m] * x[j]; } } return b; } /******************************************************************************/ void r8ge_print ( int m, int n, double a[], char *title ) /******************************************************************************/ /* Purpose: R8GE_PRINT prints an R8GE matrix. Discussion: The R8GE storage format is used for a "general" M by N matrix. A physical storage space is made for each logical entry. The two dimensional logical array is mapped to a vector, in which storage is by columns. Licensing: This code is distributed under the MIT license. Modified: 28 February 2012 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, double A[M*N], the R8GE matrix. Input, char *TITLE, a title. */ { r8ge_print_some ( m, n, a, 1, 1, m, n, title ); return; } /******************************************************************************/ void r8ge_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi, int jhi, char *title ) /******************************************************************************/ /* Purpose: R8GE_PRINT_SOME prints some of an R8GE matrix. Discussion: The R8GE storage format is used for a "general" M by N matrix. A physical storage space is made for each logical entry. The two dimensional logical array is mapped to a vector, in which storage is by columns. Licensing: This code is distributed under the MIT license. Modified: 28 February 2012 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, double A[M*N], the R8GE matrix. Input, int ILO, JLO, IHI, JHI, designate the first row and column, and the last row and column to be printed. Input, char *TITLE, a title. */ { # define INCX 5 int i; int i2hi; int i2lo; int j; int j2hi; int j2lo; printf ( "\n" ); printf ( "%s\n", title ); /* Print the columns of the matrix, in strips of 5. */ for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX ) { j2hi = j2lo + INCX - 1; j2hi = i4_min ( j2hi, n ); j2hi = i4_min ( j2hi, jhi ); printf ( "\n" ); /* For each column J in the current range... Write the header. */ printf ( " Col: " ); for ( j = j2lo; j <= j2hi; j++ ) { printf ( "%7d ", j ); } printf ( "\n" ); printf ( " Row\n" ); printf ( " ---\n" ); /* Determine the range of the rows in this strip. */ i2lo = i4_max ( ilo, 1 ); i2hi = i4_min ( ihi, m ); for ( i = i2lo; i <= i2hi; i++ ) { /* Print out (up to) 5 entries in row I, that lie in the current strip. */ printf ( "%5d ", i ); for ( j = j2lo; j <= j2hi; j++ ) { printf ( "%12g ", a[i-1+(j-1)*m] ); } printf ( "\n" ); } } return; # undef INCX } /******************************************************************************/ double r8vec_dot_product ( int n, double a1[], double a2[] ) /******************************************************************************/ /* Purpose: R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's. Licensing: This code is distributed under the MIT license. Modified: 26 July 2007 Author: John Burkardt Parameters: Input, int N, the number of entries in the vectors. Input, double A1[N], A2[N], the two vectors to be considered. Output, double R8VEC_DOT_PRODUCT, the dot product of the vectors. */ { int i; double value; value = 0.0; for ( i = 0; i < n; i++ ) { value = value + a1[i] * a2[i]; } return value; } /******************************************************************************/ double *r8vec_indicator1_new ( int n ) /******************************************************************************/ /* Purpose: R8VEC_INDICATOR1_NEW sets an R8VEC to the indicator1 vector {1,2,3...}. Licensing: This code is distributed under the MIT license. Modified: 26 August 2008 Author: John Burkardt Parameters: Input, int N, the number of elements of A. Output, double R8VEC_INDICATOR1_NEW[N], the array. */ { double *a; int i; a = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i <= n-1; i++ ) { a[i] = ( double ) ( i + 1 ); } return a; } /******************************************************************************/ double r8vec_norm ( int n, double a[] ) /******************************************************************************/ /* Purpose: R8VEC_NORM returns the L2 norm of an R8VEC. Discussion: The vector L2 norm is defined as: R8VEC_NORM = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). Licensing: This code is distributed under the MIT license. Modified: 01 March 2003 Author: John Burkardt Parameters: Input, int N, the number of entries in A. Input, double A[N], the vector whose L2 norm is desired. Output, double R8VEC_NORM, the L2 norm of A. */ { int i; double v; v = 0.0; for ( i = 0; i < n; i++ ) { v = v + a[i] * a[i]; } v = sqrt ( v ); return v; } /******************************************************************************/ double r8vec_norm_affine ( int n, double v0[], double v1[] ) /******************************************************************************/ /* Purpose: R8VEC_NORM_AFFINE returns the affine L2 norm of an R8VEC. Discussion: The affine vector L2 norm is defined as: R8VEC_NORM_AFFINE(V0,V1) = sqrt ( sum ( 1 <= I <= N ) ( V1(I) - V0(I) )^2 ) Licensing: This code is distributed under the MIT license. Modified: 27 October 2010 Author: John Burkardt Parameters: Input, int N, the dimension of the vectors. Input, double V0[N], the base vector. Input, double V1[N], the vector whose affine L2 norm is desired. Output, double R8VEC_NORM_AFFINE, the affine L2 norm of V1. */ { int i; double value; value = 0.0; for ( i = 0; i < n; i++ ) { value = value + ( v1[i] - v0[i] ) * ( v1[i] - v0[i] ); } value = sqrt ( value ); return value; } /******************************************************************************/ void r8vec_print ( int n, double a[], char *title ) /******************************************************************************/ /* Purpose: R8VEC_PRINT prints an R8VEC. Discussion: An R8VEC is a vector of R8's. Licensing: This code is distributed under the MIT license. Modified: 08 April 2009 Author: John Burkardt Parameters: Input, int N, the number of components of the vector. Input, double A[N], the vector to be printed. Input, char *TITLE, a title. */ { int i; printf ( "\n" ); printf ( "%s\n", title ); printf ( "\n" ); for ( i = 0; i < n; i++ ) { printf ( " %8d %14f\n", i, a[i] ); } return; } /******************************************************************************/ double *r8vec_uniform_01_new ( int n, int *seed ) /******************************************************************************/ /* Purpose: R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC. Discussion: This routine implements the recursion seed = 16807 * seed mod ( 2^31 - 1 ) unif = seed / ( 2^31 - 1 ) The integer arithmetic never requires more than 32 bits, including a sign bit. Licensing: This code is distributed under the MIT license. Modified: 19 August 2004 Author: John Burkardt Reference: Paul Bratley, Bennett Fox, Linus Schrage, A Guide to Simulation, Second Edition, Springer, 1987, ISBN: 0387964673, LC: QA76.9.C65.B73. Bennett Fox, Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators, ACM Transactions on Mathematical Software, Volume 12, Number 4, December 1986, pages 362-376. Pierre L'Ecuyer, Random Number Generation, in Handbook of Simulation, edited by Jerry Banks, Wiley, 1998, ISBN: 0471134031, LC: T57.62.H37. Peter Lewis, Allen Goodman, James Miller, A Pseudo-Random Number Generator for the System/360, IBM Systems Journal, Volume 8, Number 2, 1969, pages 136-143. Parameters: Input, int N, the number of entries in the vector. Input/output, int *SEED, a seed for the random number generator. Output, double R8VEC_UNIFORM_01_NEW[N], the vector of pseudorandom values. */ { int i; int i4_huge = 2147483647; int k; double *r; if ( *seed == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R8VEC_UNIFORM_01_NEW - Fatal error!\n" ); fprintf ( stderr, " Input value of SEED = 0.\n" ); exit ( 1 ); } r = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { k = *seed / 127773; *seed = 16807 * ( *seed - k * 127773 ) - k * 2836; if ( *seed < 0 ) { *seed = *seed + i4_huge; } r[i] = ( double ) ( *seed ) * 4.656612875E-10; } return r; } /******************************************************************************/ void r8vec2_print ( int n, double a1[], double a2[], char *title ) /******************************************************************************/ /* Purpose: R8VEC2_PRINT prints an R8VEC2. Discussion: An R8VEC2 is a dataset consisting of N pairs of real values, stored as two separate vectors A1 and A2. Licensing: This code is distributed under the MIT license. Modified: 26 March 2009 Author: John Burkardt Parameters: Input, int N, the number of components of the vector. Input, double A1[N], double A2[N], the vectors to be printed. Input, char *TITLE, a title. */ { int i; fprintf ( stdout, "\n" ); fprintf ( stdout, "%s\n", title ); fprintf ( stdout, "\n" ); for ( i = 0; i < n; i++ ) { fprintf ( stdout, " %4d: %14g %14g\n", i, a1[i], a2[i] ); } return; }