# include # include # include # include # include # include "r8pbl.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 GNU LGPL 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 GNU LGPL 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 GNU LGPL 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 GNU LGPL 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 GNU LGPL 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; if ( *seed == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R8_UNIFORM_01 - Fatal error!\n" ); fprintf ( stderr, " Input value of SEED = 0\n" ); exit ( 1 ); } 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 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 GNU LGPL 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 GNU LGPL 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 *r8pbl_dif2 ( int n, int ml ) /******************************************************************************/ /* Purpose: R8PBL_DIF2 returns the DIF2 matrix in R8PBL format. Example: N = 5 2 -1 . . . -1 2 -1 . . . -1 2 -1 . . . -1 2 -1 . . . -1 2 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 GNU LGPL license. Modified: 21 July 2016 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 N, the number of rows and columns. Input, int ML, the number of subdiagonals. ML must be at least 0, and no more than N-1. Output, double R8PBL_DIF2[(ML+1)*N], the matrix. */ { double *a; int j; a = r8vec_zeros_new ( ( ml + 1 ) * n ); for ( j = 0; j < n; j++ ) { a[0+j*(ml+1)] = 2.0; } for ( j = 0; j < n - 1; j++ ) { a[1+j*(ml+1)] = -1.0; } return a; } /******************************************************************************/ double *r8pbl_indicator ( int n, int ml ) /******************************************************************************/ /* Purpose: R8PBL_INDICATOR sets up an R8PBL indicator matrix. Discussion: The R8PBL storage format is used for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. Licensing: This code is distributed under the GNU LGPL license. Modified: 14 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int ML, the number of subdiagonals in the matrix. ML must be at least 0 and no more than N-1. Output, double R8PBL_INDICATOR[(ML+1)*N], the R8PBL matrix. */ { double *a; int fac; int i; int j; a = r8vec_zeros_new ( ( ml + 1 ) * n ); fac = i4_power ( 10, i4_log_10 ( n ) + 1 ); /* Set the meaningful values. */ for ( i = 0; i <= n; i++ ) { for ( j = i4_max ( 1, i - ml ); j <= i; j++ ) { a[i-j+(j-1)*(ml+1)] = ( double ) ( fac * i + j ); } } return a; } /******************************************************************************/ double *r8pbl_mv ( int n, int ml, double a[], double x[] ) /******************************************************************************/ /* Purpose: R8PBL_MV multiplies an R8PBL matrix by an R8VEC. Discussion: The R8PBL storage format is for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. Licensing: This code is distributed under the GNU LGPL license. Modified: 21 July 2016 Author: John Burkardt Parameters: Input, int N, the order of the matrix. Input, int ML, the number of subdiagonals in the matrix. ML must be at least 0 and no more than N-1. Input, double A([ML+1)*N], the R8PBL matrix. Input, double X[N], the vector to be multiplied by A. Output, double R8PBL_MV[M], the result vector A * x. */ { double aij; double *b; int i; int j; int k; b = r8vec_zeros_new ( n ); /* Multiply X by the diagonal of the matrix. */ for ( j = 0; j < n; j++ ) { b[j] = a[0+j*(ml+1)] * x[j]; } /* Multiply X by the subdiagonals of the matrix. */ for ( k = 0; k < ml; k++ ) { for ( j = 0; j < n - k; j++ ) { i = j + k; aij = a[k+1+j*(ml+1)]; b[i] = b[i] + aij * x[j]; b[j] = b[j] + aij * x[i]; } } return b; } /******************************************************************************/ void r8pbl_print ( int n, int ml, double a[], char *title ) /******************************************************************************/ /* Purpose: R8PBL_PRINT prints an R8PBL matrix. Discussion: The R8PBL storage format is used for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. Licensing: This code is distributed under the GNU LGPL license. Modified: 14 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int ML, the upper (and lower) bandwidth. ML must be nonnegative, and no greater than N-1. Input, double A[(ML+1)*N], the R8PBL matrix. Input, char *TITLE, a title. */ { r8pbl_print_some ( n, ml, a, 0, 0, n - 1, n - 1, title ); return; } /******************************************************************************/ void r8pbl_print_some ( int n, int ml, double a[], int ilo, int jlo, int ihi, int jhi, char *title ) /******************************************************************************/ /* Purpose: R8PBL_PRINT_SOME prints some of an R8PBL matrix. Discussion: The R8PBL storage format is used for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. Licensing: This code is distributed under the GNU LGPL license. Modified: 14 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int ML, the upper (and lower) bandwidth. ML must be nonnegative, and no greater than N-1. Input, double A[(ML+1)*N], the R8PBL 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 - 1 ); j2hi = i4_min ( j2hi, jhi ); printf ( "\n" ); 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, 0 ); i2lo = i4_max ( i2lo, j2lo - ml ); i2hi = i4_min ( ihi, n - 1 ); i2hi = i4_min ( i2hi, j2hi + ml ); for ( i = i2lo; i <= i2hi; i++ ) { printf ( "%4d ", i ); /* Print out (up to) 5 entries in row I, that lie in the current strip. */ for ( j = j2lo; j <= j2hi; j++ ) { if ( i <= j && j <= i + ml ) { printf ( "%12g ", a[j-i+i*(ml+1)] ); } else if ( j <= i && i <= j + ml ) { printf ( "%12g ", a[i-j+j*(ml+1)] ); } else { printf ( " " ); } } printf ( "\n" ); } } return; # undef INCX } /******************************************************************************/ double *r8pbl_random ( int n, int ml, int *seed ) /******************************************************************************/ /* Purpose: R8PBL_RANDOM randomizes an R8PBL matrix. Discussion: The R8PBL storage format is used for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. The matrix returned will be positive definite, but of limited randomness. The off diagonal elements are random values between 0 and 1, and the diagonal element of each row is selected to ensure strict diagonal dominance. Licensing: This code is distributed under the GNU LGPL license. Modified: 14 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int ML, the number of subdiagonals in the matrix. ML must be at least 0 and no more than N-1. Input/output, int *SEED, a seed for the random number generator. Output, double R8PBL_RANDOM[(ML+1)*N], the R8PBL matrix. */ { double *a; int i; int j; double r; double sum2; a = r8vec_zeros_new ( ( ml + 1 ) * n ); /* Set the off diagonal values. */ for ( i = 0; i < n; i++ ) { for ( j = i4_max ( 0, i - ml ); j <= i - 1; j++ ) { a[i-j+j*(ml+1)] = r8_uniform_01 ( seed ); } } /* Set the diagonal values. */ for ( i = 0; i < n; i++ ) { sum2 = 0.0; for ( j = i4_max ( 0, i - ml ); j <= i - 1; j++ ) { sum2 = sum2 + fabs ( a[i-j+j*(ml+1)] ); } for ( j = i + 1; j <= i4_min ( i + ml, n - 1 ); j++ ) { sum2 = sum2 + fabs ( a[j-i+i*(ml+1)] ); } r = r8_uniform_01 ( seed ); a[0+i*(ml+1)] = ( 1.0 + r ) * ( sum2 + 0.01 ); } return a; } /******************************************************************************/ double *r8pbl_to_r8ge ( int n, int ml, double a[] ) /******************************************************************************/ /* Purpose: R8PBL_TO_R8GE copies an R8PBL matrix to an R8GE matrix. Discussion: The R8PBL storage format is used for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. Licensing: This code is distributed under the GNU LGPL license. Modified: 14 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrices. N must be positive. Input, int ML, the upper bandwidth of A1. ML must be nonnegative, and no greater than N-1. Input, double A[(ML+1)*N], the R8PBL matrix. Output, double R8PBL_TO_R8GE[N*N], the R8GE matrix. */ { double *b; int i; int j; b = r8vec_zeros_new ( n * n ); for ( i = 0; i < n; i++ ) { for ( j = 0; j < n; j++ ) { if ( i <= j && j <= i + ml ) { b[i+j*n] = a[j-i+i*(ml+1)]; } else if ( i - ml <= j && j < i ) { b[i+j*n] = a[i-j+j*(ml+1)]; } } } return b; } /******************************************************************************/ double *r8pbl_zeros ( int n, int ml ) /******************************************************************************/ /* Purpose: R8PBL_ZEROS zeros an R8PBL matrix. Discussion: The R8PBL storage format is used for a symmetric positive definite band matrix. To save storage, only the diagonal and lower triangle of A is stored, in a compact diagonal format that preserves columns. The diagonal is stored in row 1 of the array. The first subdiagonal in row 2, columns 1 through ML. The second subdiagonal in row 3, columns 1 through ML-1. The ML-th subdiagonal in row ML+1, columns 1 through 1. Licensing: This code is distributed under the GNU LGPL license. Modified: 14 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int ML, the number of subdiagonals in the matrix. ML must be at least 0 and no more than N-1. Output, double R8PBL_ZEROS[(ML+1)*N], the R8PBL matrix. */ { double *a; a = r8vec_zeros_new ( ( ml + 1 ) * n ); return a; } /******************************************************************************/ 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 GNU LGPL 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 = r8vec_zeros_new ( n ); for ( i = 0; i <= n - 1; i++ ) { a[i] = ( double ) ( i + 1 ); } return a; } /******************************************************************************/ 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 GNU LGPL 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_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 GNU LGPL 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 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 GNU LGPL 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 }