# include # include # include # include # include # include "r83.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; } /******************************************************************************/ int i4_power ( int i, int j ) /******************************************************************************/ /* Purpose: I4_POWER returns the value of I^J. Licensing: This code is distributed under the MIT license. Modified: 23 October 2007 Author: John Burkardt Parameters: Input, int I, J, the base and the power. J should be nonnegative. Output, int I4_POWER, the value of I^J. */ { int k; int value; if ( j < 0 ) { if ( i == 1 ) { value = 1; } else if ( i == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "I4_POWER - Fatal error!\n" ); fprintf ( stderr, " I^J requested, with I = 0 and J negative.\n" ); exit ( 1 ); } else { value = 0; } } else if ( j == 0 ) { if ( i == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "I4_POWER - Fatal error!\n" ); fprintf ( stderr, " I^J requested, with I = 0 and J = 0.\n" ); exit ( 1 ); } else { value = 1; } } else if ( j == 1 ) { value = i; } else { value = 1; for ( k = 1; k <= j; k++ ) { value = value * i; } } 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 r83_cg ( int n, double a[], double b[], double x[] ) /******************************************************************************/ /* Purpose: R83_CG uses the conjugate gradient method on an R83 system. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). 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: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 04 June 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. Input, double A[3*N], 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 = r83_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 = r83_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 *r83_cr_fa ( int n, double a[] ) /******************************************************************************/ /* Purpose: R83_CR_FA decomposes a real tridiagonal matrix using cyclic reduction. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Once R83_CR_FA has decomposed a matrix A, then R83_CR_SL may be used to solve linear systems A * x = b. R83_CR_FA does not employ pivoting. Hence, the results can be more sensitive to ill-conditioning than standard Gauss elimination. In particular, R83_CR_FA will fail if any diagonal element of the matrix is zero. Other matrices may also cause R83_CR_FA to fail. R83_CR_FA can be guaranteed to work properly if the matrix is strictly diagonally dominant, that is, if the absolute value of the diagonal element is strictly greater than the sum of the absolute values of the offdiagonal elements, for each equation. The algorithm may be illustrated by the following figures: The initial matrix is given by: D1 U1 L1 D2 U2 L2 D3 U3 L3 D4 U4 L4 D U5 L5 D6 Rows and columns are permuted in an odd/even way to yield: D1 U1 D3 L2 U3 D5 L4 U5 L1 U2 D2 L3 U4 D4 L5 D6 A block LU decomposition is performed to yield: D1 |U1 D3 |L2 U3 D5| L4 U5 --------+-------- |D2'F3 |F1 D4'F4 | F2 D6' For large systems, this reduction is repeated on the lower right hand tridiagonal subsystem until a completely upper triangular system is obtained. The system has now been factored into the product of a lower triangular system and an upper triangular one, and the information defining this factorization may be used by R83_CR_SL to solve linear systems. Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 06 September 2015 Author: John Burkardt Reference: Roger Hockney, A fast direct solution of Poisson's equation using Fourier Analysis, Journal of the ACM, Volume 12, Number 1, pages 95-113, January 1965. Parameters: Input, int N, the order of the matrix. N must be positive. Input, double A[3*N], the R83 matrix. Output, double R83_CR_FA[3*(2*N+1)], factorization information needed by R83_CR_SL. */ { double *a_cr; int i; int iful; int ifulp; int ihaf; int il; int ilp; int inc; int incr; int ipnt; int ipntp; int j; if ( n <= 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83_CR_FA - Fatal error!\n" ); fprintf ( stderr, " Nonpositive N = %d\n", n ); exit ( 1 ); } a_cr = ( double * ) malloc ( 3 * ( 2 * n + 1 ) * sizeof ( double ) ); for ( j = 0; j < 2 * n + 1; j++ ) { for ( i = 0; i < 3; i++ ) { a_cr[i+j*3] = 0.0; } } if ( n == 1 ) { a_cr[1+0*3] = 1.0 / a[1+0*3]; return a_cr; } for ( j = 1; j <= n-1; j++ ) { a_cr[0+j*3] = a[0+j*3]; } for ( j = 1; j <= n; j++ ) { a_cr[1+j*3] = a[1+(j-1)*3]; } for ( j = 1; j <= n-1; j++ ) { a_cr[2+j*3] = a[2+(j-1)*3]; } il = n; ipntp = 0; while ( 1 < il ) { ipnt = ipntp; ipntp = ipntp + il; if ( ( il % 2 ) == 1 ) { inc = il + 1; } else { inc = il; } incr = inc / 2; il = il / 2; ihaf = ipntp + incr + 1; ifulp = ipnt + inc + 2; for ( ilp = incr; 1 <= ilp; ilp-- ) { ifulp = ifulp - 2; iful = ifulp - 1; ihaf = ihaf - 1; a_cr[1+iful*3] = 1.0 / a_cr[1+iful*3]; a_cr[2+iful*3] = a_cr[2+iful*3] * a_cr[1+iful*3]; a_cr[0+ifulp*3] = a_cr[0+ifulp*3] * a_cr[1+(ifulp+1)*3]; a_cr[1+ihaf*3] = a_cr[1+ifulp*3] - a_cr[0+iful*3] * a_cr[2+iful*3] - a_cr[0+ifulp*3] * a_cr[2+ifulp*3]; a_cr[2+ihaf*3] = -a_cr[2+ifulp*3] * a_cr[2+(ifulp+1)*3]; a_cr[0+ihaf*3] = -a_cr[0+ifulp*3] * a_cr[0+(ifulp+1)*3]; } } a_cr[1+(ipntp+1)*3] = 1.0 / a_cr[1+(ipntp+1)*3]; return a_cr; } /******************************************************************************/ double *r83_cr_sl ( int n, double a_cr[], double b[] ) /******************************************************************************/ /* Purpose: R83_CR_SL solves a real linear system factored by R83_CR_FA. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). The matrix A must be tridiagonal. R83_CR_FA is called to compute the LU factors of A. It does so using a form of cyclic reduction. If the factors computed by R83_CR_FA are passed to R83_CR_SL, then one or many linear systems involving the matrix A may be solved. Note that R83_CR_FA does not perform pivoting, and so the solution produced by R83_CR_SL may be less accurate than a solution produced by a standard Gauss algorithm. However, such problems can be guaranteed not to occur if the matrix A is strictly diagonally dominant, that is, if the absolute value of the diagonal coefficient is greater than the sum of the absolute values of the two off diagonal coefficients, for each row of the matrix. Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 30 May 2009 Author: C version by John Burkardt Reference: Roger Hockney, A fast direct solution of Poisson's equation using Fourier Analysis, Journal of the ACM, Volume 12, Number 1, pages 95-113, January 1965. Parameters: Input, int N, the order of the matrix. N must be positive. Input, double A_CR[3*(2*N+1)], factorization information computed by R83_CR_FA. Input, double B[N], the right hand side. Output, double R83_CR_SL[N], the solution. */ { int i; int iful; int ifulm; int ihaf; int il; int ipnt; int ipntp; int ndiv; double *rhs; double *x; if ( n <= 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83_CR_SL - Fatal error!\n" ); fprintf ( stderr, " Nonpositive N = %d\n", n ); exit ( 1 ); } if ( n == 1 ) { x = ( double * ) malloc ( 1 * sizeof ( double ) ); x[0] = a_cr[1+1*3] * b[0]; return x; } /* Set up RHS. */ rhs = ( double * ) malloc ( ( 2 * n + 1 ) * sizeof ( double ) ); rhs[0] = 0.0; for ( i = 1; i <= n; i++ ) { rhs[i] = b[i-1]; } for ( i = n + 1; i <= 2 * n; i++ ) { rhs[i] = 0.0; } il = n; ndiv = 1; ipntp = 0; while ( 1 < il ) { ipnt = ipntp; ipntp = ipntp + il; il = il / 2; ndiv = ndiv * 2; ihaf = ipntp; for ( iful = ipnt + 2; iful <= ipntp; iful = iful + 2 ) { ihaf = ihaf + 1; rhs[ihaf] = rhs[iful] - a_cr[2+(iful-1)*3] * rhs[iful-1] - a_cr[0+iful*3] * rhs[iful+1]; } } rhs[ihaf] = rhs[ihaf] * a_cr[1+ihaf*3]; ipnt = ipntp; while ( 0 < ipnt ) { ipntp = ipnt; ndiv = ndiv / 2; il = n / ndiv; ipnt = ipnt - il; ihaf = ipntp; for ( ifulm = ipnt + 1; ifulm <= ipntp; ifulm = ifulm + 2 ) { iful = ifulm + 1; ihaf = ihaf + 1; rhs[iful] = rhs[ihaf]; rhs[ifulm] = a_cr[1+ifulm*3] * ( rhs[ifulm] - a_cr[2+(ifulm-1)*3] * rhs[ifulm-1] - a_cr[0+ifulm*3] * rhs[iful] ); } } x = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { x[i] = rhs[i+1]; } free ( rhs ); return x; } /******************************************************************************/ double *r83_cr_sls ( int n, double a_cr[], int nb, double b[] ) /******************************************************************************/ /* Purpose: R83_CR_SLS solves several real linear systems factored by R83_CR_FA. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). The matrix A must be tridiagonal. R83_CR_FA is called to compute the LU factors of A. It does so using a form of cyclic reduction. If the factors computed by R83_CR_FA are passed to R83_CR_SLS, then one or many linear systems involving the matrix A may be solved. Note that R83_CR_FA does not perform pivoting, and so the solutions produced by R83_CR_SLS may be less accurate than a solution produced by a standard Gauss algorithm. However, such problems can be guaranteed not to occur if the matrix A is strictly diagonally dominant, that is, if the absolute value of the diagonal coefficient is greater than the sum of the absolute values of the two off diagonal coefficients, for each row of the matrix. Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 09 May 2010 Author: John Burkardt Reference: Roger Hockney, A fast direct solution of Poisson's equation using Fourier Analysis, Journal of the ACM, Volume 12, Number 1, pages 95-113, January 1965. Parameters: Input, int N, the order of the matrix. N must be positive. Input, double A_CR[3*(2*N+1)], factorization information computed by R83_CR_FA. Input, int NB, the number of systems. Input, double B[N*NB], the right hand sides. Output, double R83_CR_SL[N*NB], the solutions. */ { int i; int iful; int ifulm; int ihaf; int il; int ipnt; int ipntp; int j; int ndiv; double *rhs; double *x; if ( n <= 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83_CR_SLS - Fatal error!\n" ); fprintf ( stderr, " Nonpositive N = %d\n", n ); exit ( 1 ); } if ( n == 1 ) { x = ( double * ) malloc ( n * nb * sizeof ( double ) ); for ( j = 0; j < nb; j++ ) { x[0+j*n] = a_cr[1+0*3] * b[0+j*n]; } return x; } // // Set up RHS. // rhs = ( double * ) malloc ( ( 2 * n + 1 ) * nb * sizeof ( double ) ); for ( j = 0; j < nb; j++ ) { rhs[0+j*(2*n+1)] = 0.0; for ( i = 1; i <= n; i++ ) { rhs[i+j*(2*n+1)] = b[i-1+j*n]; } for ( i = n + 1; i <= 2 * n; i++ ) { rhs[i+j*(2*n+1)] = 0.0; } } il = n; ndiv = 1; ipntp = 0; while ( 1 < il ) { ipnt = ipntp; ipntp = ipntp + il; il = il / 2; ndiv = ndiv * 2; for ( j = 0; j < nb; j++ ) { ihaf = ipntp; for ( iful = ipnt + 2; iful <= ipntp; iful = iful + 2 ) { ihaf = ihaf + 1; rhs[ihaf+j*(2*n+1)] = rhs[iful+j*(2*n+1)] - a_cr[2+(iful-1)*3] * rhs[iful-1+j*(2*n+1)] - a_cr[0+iful*3] * rhs[iful+1+j*(2*n+1)]; } } } for ( j = 0; j < nb; j++ ) { rhs[ihaf+j*(2*n+1)] = rhs[ihaf+j*(2*n+1)] * a_cr[1+ihaf*3]; } ipnt = ipntp; while ( 0 < ipnt ) { ipntp = ipnt; ndiv = ndiv / 2; il = n / ndiv; ipnt = ipnt - il; for ( j = 0; j < nb; j++ ) { ihaf = ipntp; for ( ifulm = ipnt + 1; ifulm <= ipntp; ifulm = ifulm + 2 ) { iful = ifulm + 1; ihaf = ihaf + 1; rhs[iful+j*(2*n+1)] = rhs[ihaf+j*(2*n+1)]; rhs[ifulm+j*(2*n+1)] = a_cr[1+ifulm*3] * ( rhs[ifulm+j*(2*n+1)] - a_cr[2+(ifulm-1)*3] * rhs[ifulm-1+j*(2*n+1)] - a_cr[0+ifulm*3] * rhs[iful+j*(2*n+1)] ); } } } x = ( double * ) malloc ( n * nb * sizeof ( double ) ); for ( j = 0; j < nb; j++ ) { for ( i = 0; i < n; i++ ) { x[i+j*n] = rhs[i+1+j*(2*n+1)]; } } free ( rhs ); return x; } /******************************************************************************/ double *r83_dif2 ( int m, int n ) /******************************************************************************/ /* Purpose: R83_DIF2 returns the DIF2 matrix in R83 format. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 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: 01 September 2015 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*N], the matrix. */ { double *a; int i; int i_hi; int i_lo; int j; a = ( double * ) malloc ( 3 * n * sizeof ( double ) ); for ( j = 0; j < n; j++ ) { for ( i = 0; i < 3; i++ ) { a[i+j*3] = 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++ ) { if ( i == j - 1 ) { a[i-j+1+j*3] = -1.0; } else if ( i == j ) { a[i-j+1+j*3] = +2.0; } else if ( i == j + 1 ) { a[i-j+1+j*3] = -1.0; } } } return a; } /******************************************************************************/ void r83_gs_sl ( int n, double a[], double b[], double x[], int it_max ) /******************************************************************************/ /* Purpose: R83_GS_SL solves an R83 system using Gauss-Seidel iteration. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). 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: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 06 September 2015 Author: John Burkardt Parameters: Input, int N, the order of the matrix. Input, double A[3*N], the R83 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. */ for ( i = 0; i < n; i++ ) { if ( a[1+i*3] == 0.0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83_GS_SL - Fatal error!\n" ); fprintf ( stderr, " Zero diagonal entry, index = %d\n", i ); exit ( 1 ); } } for ( it_num = 1; it_num <= it_max; it_num++ ) { x[0] = ( b[0] - a[0+1*3] * x[1] ) / a[1+0*3]; for ( i = 1; i < n - 1; i++ ) { x[i] = ( b[i] - a[2+(i-1)*3] * x[i-1] - a[0+(i+1)*3] * x[i+1] ) / a[1+i*3]; } x[n-1] = ( b[n-1] - a[2+(n-2)*3] * x[n-2] ) / a[1+(n-1)*3]; } return; } /******************************************************************************/ double *r83_indicator ( int m, int n ) /******************************************************************************/ /* Purpose: R83_INDICATOR sets up an R83 indicator matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Output, double R83_INDICATOR[3*N], the R83 indicator matrix. */ { double *a; int fac; int i; int i_hi; int i_lo; int j; a = ( double * ) malloc ( 3 * n * sizeof ( double ) ); fac = i4_power ( 10, i4_log_10 ( n ) + 1 ); for ( j = 0; j < n; j++ ) { for ( i = 0; i < 3; i++ ) { a[i+j*3] = 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++ ) { a[i-j+1+j*3] = ( double ) ( fac * ( i + 1 ) + j + 1 ); } } return a; } /******************************************************************************/ void r83_jac_sl ( int n, double a[], double b[], double x[], int it_max ) /******************************************************************************/ /* Purpose: R83_JAC_SL solves an R83 system using Jacobi iteration. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). 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: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 06 September 2015 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be at least 2. Input, double A[3*N], the R83 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. */ for ( i = 0; i < n; i++ ) { if ( a[1+i*3] == 0.0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83_JAC_SL - Fatal error!\n" ); fprintf ( stderr, " Zero diagonal entry, index = %d\n", i ); exit ( 1 ); } } for ( it_num = 1; it_num <= it_max; it_num++ ) { /* Solve A*x=b: */ xnew[0] = b[0] - a[0+1*3] * x[1]; for ( i = 1; i < n - 1; i++ ) { xnew[i] = b[i] - a[2+(i-1)*3] * x[i-1] - a[0+(i+1)*3] * x[i+1]; } xnew[n-1] = b[n-1] - a[2+(n-2)*3] * x[n-2]; /* Divide by the diagonal term, and overwrite X. */ for ( i = 0; i < n; i++ ) { x[i] = xnew[i] / a[1+i*3]; } } free ( xnew ); return; } /******************************************************************************/ double *r83_mtv ( int m, int n, double a[], double x[] ) /******************************************************************************/ /* Purpose: R83_MTV multiplies a vector times an R83 matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the linear system. Input, double A[3*N], the R83 matrix. Input, double X[M], the vector to be multiplied. Output, double R83_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+j*3]; } } return b; } /******************************************************************************/ double *r83_mv ( int m, int n, double a[], double x[] ) /******************************************************************************/ /* Purpose: R83_MV multiplies an R83 matrix times a vector. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the linear system. Input, double A[3*N], the R83 matrix. Input, double X[N], the vector to be multiplied by A. Output, double R83_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+j*3] * x[j]; } } return b; } /******************************************************************************/ void r83_print ( int m, int n, double a[], char *title ) /******************************************************************************/ /* Purpose: R83_PRINT prints an R83 matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3*N], the R83 matrix. Input, char *TITLE, a title. */ { r83_print_some ( m, n, a, 1, 1, m, n, title ); return; } /******************************************************************************/ void r83_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi, int jhi, char *title ) /******************************************************************************/ /* Purpose: R83_PRINT_SOME prints some of an R83 matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 August 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3*N], the R83 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+(j-1)*3] ); } } printf ( "\n" ); } } return; # undef INCX } /******************************************************************************/ double *r83_random ( int m, int n, int *seed ) /******************************************************************************/ /* Purpose: R83_RANDOM randomizes an R83 matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 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 R83_RANDOM[3*N], the R83 matrix. */ { double *a; int i; int i_hi; int i_lo; int j; a = ( double * ) malloc ( 3 * n * sizeof ( double ) ); for ( j = 0; j < n; j++ ) { for ( i = 0; i < 3; i++ ) { a[i+j*3] = 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++ ) { a[i-j+1+j*3] = r8_uniform_01 ( seed ); } } return a; } /******************************************************************************/ double *r83_res ( int m, int n, double a[], double x[], double b[] ) /******************************************************************************/ /* Purpose: R83_RES computes the residual R = B-A*X for R83 matrices. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 04 June 2014 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[3*N], the matrix. Input, double X[N], the vector to be multiplied by A. Input, double B[M], the desired result A * x. Output, double R83_RES[M], the residual R = B - A * X. */ { int i; double *r; r = r83_mv ( m, n, a, x ); for ( i = 0; i < m; i++ ) { r[i] = b[i] - r[i]; } return r; } /******************************************************************************/ double *r83_to_r8ge ( int m, int n, double a[] ) /******************************************************************************/ /* Purpose: R83_TO_R8GE copies an R83 matrix to an R8GE matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Input, double A[3*N], the R83 matrix. Output, double R83_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+j*3]; } } return b; } /******************************************************************************/ double *r83_zeros ( int m, int n ) /******************************************************************************/ /* Purpose: R83_ZEROS zeros an R83 matrix. Discussion: The R83 storage format is used for a tridiagonal matrix. The superdiagonal is stored in entries (1,2:min(M+1,N)). The diagonal in entries (2,1:min(M,N)). The subdiagonal in (3,min(M-1,N)). R8GE A(I,J) = R83 A[I-J+1+J*3] (0 based indexing). Example: An R83 matrix of order 3x5 would be stored: * A12 A23 A34 * A11 A22 A33 * * A21 A32 * * * An R83 matrix of order 5x5 would be stored: * A12 A23 A34 A45 A11 A22 A33 A44 A55 A21 A32 A43 A54 * An R83 matrix of order 5x3 would be stored: * A12 A23 A11 A22 A33 A21 A32 A43 Licensing: This code is distributed under the MIT license. Modified: 01 September 2015 Author: John Burkardt Parameters: Input, int M, N, the order of the matrix. Output, double R83_ZEROS[3*N], the R83 matrix. */ { double *a; int i; int j; a = ( double * ) malloc ( 3 * n * sizeof ( double ) ); for ( j = 0; j < n; j++ ) { for ( i = 0; i < 3; i++ ) { a[i+j*3] = 0.0; } } return a; } /******************************************************************************/ double *r83np_fs ( int n, double a[], double b[] ) /******************************************************************************/ /* Purpose: R83NP_FS factors and solves an R83NP system. Discussion: The R83NP storage format is used for a tridiagonal matrix. The subdiagonal is in entries (0,1:N-1), the diagonal is in entries (1,0:N-1), the superdiagonal is in entries (2,0:N-2). This algorithm requires that each diagonal entry be nonzero. It does not use pivoting, and so can fail on systems that are actually nonsingular. The "R83NP" format used for this routine is different from the R83 format. Here, we insist that the nonzero entries for a given row now appear in the corresponding column of the packed array. Example: Here is how an R83 matrix of order 5 would be stored: * A21 A32 A43 A54 A11 A22 A33 A44 A55 A12 A23 A34 A45 * Licensing: This code is distributed under the MIT license. Modified: 17 May 2009 Author: John Burkardt Parameters: Input, int N, the order of the linear system. Input/output, double A[3*N]. On input, the nonzero diagonals of the linear system. On output, the data in these vectors has been overwritten by factorization information. Input, double B[N], the right hand side. Output, double R83NP_FS[N], the solution of the linear system. */ { int i; double *x; /* Check. */ for ( i = 0; i < n; i++ ) { if ( a[1+i*3] == 0.0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R83NP_FS - Fatal error!\n" ); fprintf ( stderr, " A[1+%d*3] = 0.\n", i ); exit ( 1 ); } } x = r8vec_zeros_new ( n ); for ( i = 0; i < n; i++ ) { x[i] = b[i]; } for ( i = 1; i < n; i++ ) { a[1+i*3] = a[1+i*3] - a[2+(i-1)*3] * a[0+i*3] / a[1+(i-1)*3]; x[i] = x[i] - x[i-1] * a[0+i*3] / a[1+(i-1)*3]; } x[n-1] = x[n-1] / a[1+(n-1)*3]; for ( i = n - 2; 0 <= i; i-- ) { x[i] = ( x[i] - a[2+i*3] * x[i+1] ) / a[1+i*3]; } return x; } /******************************************************************************/ 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; } /******************************************************************************/ double *r8vec_zeros_new ( int n ) /******************************************************************************/ /* Purpose: R8VEC_ZEROS_NEW creates and zeroes an R8VEC. Discussion: An R8VEC is a vector of R8's. Licensing: This code is distributed under the MIT license. Modified: 25 March 2009 Author: John Burkardt Parameters: Input, int N, the number of entries in the vector. Output, double R8VEC_ZEROS_NEW[N], a vector of zeroes. */ { double *a; int i; a = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { a[i] = 0.0; } return a; } /******************************************************************************/ 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; } /******************************************************************************/ void timestamp ( ) /******************************************************************************/ /* Purpose: TIMESTAMP prints the current YMDHMS date as a time stamp. Example: 31 May 2001 09:45:54 AM Licensing: This code is distributed under the MIT license. Modified: 24 September 2003 Author: John Burkardt Parameters: None */ { # define TIME_SIZE 40 static char time_buffer[TIME_SIZE]; const struct tm *tm; time_t now; now = time ( NULL ); tm = localtime ( &now ); strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm ); printf ( "%s\n", time_buffer ); return; # undef TIME_SIZE }