double angle_shift ( double alpha, double beta );
double angle_shift_deg ( double alpha, double beta );
double *angle_to_rgb ( double angle );
void axis_limits ( double xmin, double xmax, int ndivs, double *pxmin, 
  double *pxmax, double *pxdiv, int *nticks );
int bar_check ( int digit[12] );
char *bar_code ( int digit[] );
char *bar_digit_code_left ( int digit );
char *bar_digit_code_right ( int digit );
void bin_to_d_even ( int nbin, int bin, double a, double b, double *cmin, 
  double *cmax );
double bmi_english ( double w_lb, double h_ft, double h_in );
double bmi_metric ( double w_kg, double h_m );
float c4_abs ( complex <float> x );
float c4_argument ( complex <float> x );
float c4_magnitude ( complex <float> x );
void c4mat_print_some ( int m, int n, complex <float> a[], int ilo, int jlo, 
  int ihi, int jhi, char *title );
void c4vec_print_some ( int n, complex <float> a[], int i_lo, int i_hi, 
  char *title );
double c8_argument ( complex <double> x );
complex <double> c8_cube_root ( complex <double> x );
complex <double> c8_i ( );
bool c8_le_l1 ( complex <double> x, complex <double> y );
bool c8_le_l2 ( complex <double> x, complex <double> y );
bool c8_le_li ( complex <double> x, complex <double> y );
double c8_magnitude ( complex <double> x );
double c8_norm_l1 ( complex <double> x );
double c8_norm_l2 ( complex <double> x );
double c8_norm_li ( complex <double> x );
complex <double> c8_normal_01 ( int *seed );
complex <double> c8_one ( );;
void c8_print ( complex <double> a, char *title );
complex <double> c8_sqrt ( complex <double> x );
void c8_swap ( complex <double> *x, complex <double> *y );
complex <double> c8_uniform_01 ( int *seed );
complex <double> c8_zero ( );
void c8mat_copy ( int m, int n, complex <double> a1[], complex <double> a2[] );
complex <double> *c8mat_copy_new ( int m, int n, complex <double> a1[] );
complex <double> *c8mat_identity ( int n );
complex <double> *c8mat_indicator ( int m, int n );
void c8mat_nint ( int m, int n, complex <double> a[] );
void c8mat_print ( int m, int n, complex <double> a[], char *title );
void c8mat_print_some ( int m, int n, complex <double> a[], int ilo, int jlo,
  int ihi, int jhi, char *title );
complex <double> *c8mat_uniform_01 ( int m, int n, int *seed );
void c8vec_copy ( int n, complex <double> a1[], complex <double> a2[] );
complex <double> *c8vec_copy_new ( int n, complex <double> a1[] );
complex <double> *c8vec_indicator ( int n );
double c8vec_norm_l2 ( int n, complex <double> a[] );
void c8vec_print ( int n, complex <double> a[], char *title );
complex <double> *c8vec_uniform_01 ( int n, int *seed );
complex <double> *c8vec_unity ( int n );
char ch_cap ( char c );
bool ch_eqi ( char ch1, char ch2 );
bool ch_is_digit ( char c );
int ch_to_digit ( char ch );
double degrees_to_radians ( double degrees );
double e_constant ( );
double euler_constant ( );
void fac_div ( int prime_num, int npower1[], int npower2[], int npower3[] );
void fac_gcd ( int prime_num, int npower1[], int npower2[], int npower3[] );
void fac_lcm ( int prime_num, int npower1[], int npower2[], int npower3[] );
void fac_mul ( int prime_num, int npower1[], int npower2[], int npower3[] );
void fac_print ( int prime_num, int npower[] );
int fac_to_i4 ( int prime_num, int npower[] );
void fac_to_rat ( int prime_num, int npower[], int *top, int *bot );
double feet_to_meters ( double ft );
double gauss_sum ( int dim_num, int n, double amplitude[], double center[], 
  double width[], double x[] );
unsigned long get_seed ( );
double *grid1 ( int dim_num, int nstep, double x1[], double x2[] );
double *grid1n ( int j, int dim_num, int nstep, double x1[], double x2[] );
double *grid2 ( int j1, int j2, int dim_num, int nstep, double x1[], 
  double x2[] );
double *grid2n ( int j, int j1, int j2, int dim_num, double x1[], double x2[] );
double *grid3 ( int dim_num, int nstep1, int nstep2, double x1[], double x2[],
  double x3[] );
double *grid3n ( int j, int k, int dim_num, int nstep1, int nstep2, 
  double x1[], double x2[], double x3[] );
double *grid4 ( int j1, int j2, int k1, int k2, int dim_num, int nstep1, 
  int nstep2, double x1[], double x2[], double x3[] );
double *grid4n ( int j, int j1, int j2, int k, int k1, int k2, int dim_num, 
  int nstep1, int nstep2, double x1[], double x2[], double x3[] );
short int i2_reverse_bytes ( short int x );
int i4_bit_hi1 ( int n );
int i4_bit_lo0 ( int n );
int i4_bit_lo1 ( int n );
int i4_bit_reverse ( int i, int n );
int i4_characteristic ( int q );
int i4_choose ( int n, int k );
int i4_div_rounded ( int a, int b );
int i4_divp ( int i, int j );
bool i4_even ( int i );
int i4_factorial ( int n );
int i4_gcd ( int i, int j );
int i4_gcdb ( int i, int j, int k );
int i4_huge ( );
double i4_huge_normalizer ( );
bool i4_is_power_of_2 ( int n );
bool i4_is_prime ( int n );
int i4_lcm ( int i, int j );
int i4_log_2 ( int i );
int i4_log_10 ( int i );
int i4_log_i4 ( int i4, int j4 );
int i4_log_r8 ( int x, double b );
void i4_mant ( double x, int *s, int *j, int *k, int *l );
int i4_max ( int i1, int i2 );
int i4_min ( int i1, int i2 );
void i4_moddiv ( int n, int d, int *m, int *r );
int i4_modp ( int i, int j );
int i4_mop ( int i );
bool i4_odd ( int i );
int i4_power ( int i, int j );
int i4_reverse_bytes ( int x );
int i4_sign ( int i );
void i4_swap ( int *i, int *j );
void i4_swap3 ( int *i, int *j, int *k );
double i4_to_angle ( int i );
int *i4_to_digits_decimal ( int i, int n );
int *i4_to_fac ( int i, int nprime );
void i4_to_halton ( int dim_num, int step, int seed[], int leap[], int base[], 
  double r[] );
char i4_to_isbn ( int i );
int i4_uniform ( int b, int c, int *seed );;
void i4_unswap3 ( int *i, int *j, int *k );
int i4_walsh_1d ( double x, int digit );
int i4_width ( int i );
int i4_wrap ( int ival, int ilo, int ihi );
unsigned int i4_xor ( unsigned int i, unsigned int j );
int i4col_compare ( int m, int n, int a[], int i, int j );
int i4col_find ( int m, int n, int a[], int ivec[] );
void i4col_find_item ( int m, int n, int a[], int item, int *row, int *col );
void i4col_find_pair_wrap ( int m, int n, int a[], int item1, int item2,
  int *row, int *col );
int *i4col_first_index ( int m, int n, int a[] );
void i4col_sort_a ( int m, int n, int a[] );
void i4col_sort_d ( int m, int n, int a[] );
void i4col_sort2_a ( int m, int n, int a[] );
void i4col_sort2_d ( int m, int n, int a[] );
int i4col_sorted_singleton_count ( int m, int n, int a[] );
void i4col_sorted_unique ( int m, int n, int a[], int *unique_num );
int i4col_sorted_unique_count ( int m, int n, int a[] );
void i4col_swap ( int m, int n, int a[], int icol1, int icol2 );
int *i4col_unique_index ( int m, int n, int a[] );
double i4int_to_r8int ( int imin, int imax, int i, double rmin, double rmax );
void i4mat_copy ( int m, int n, int a1[], int a2[] );
int *i4mat_copy_new ( int m, int n, int a1[] );
void i4mat_elim ( int m, int n, int a[] );
void i4mat_flip_rows ( int m, int n, int a[] );
int *i4mat_indicator ( int m, int n );
int *i4mat_l1_inverse ( int n, int a[] );
void i4mat_max_index ( int m, int n, int a[], int *i_max, int *j_max );
void i4mat_min_index ( int m, int n, int a[], int *i_min, int *j_min );
int *i4mat_mm ( int n1, int n2, int n3, int a[], int b[] );
void i4mat_perm_uniform ( int n, int a[], int *seed );
void i4mat_perm2_uniform ( int m, int n, int a[], int *seed );
void i4mat_print ( int m, int n, int a[], char *title );
void i4mat_print_some ( int m, int n, int a[], int ilo, int jlo, int ihi, 
  int jhi, char *title );
void i4mat_red ( int m, int n, int a[], int row[], int col[] );
void i4mat_transpose_print ( int m, int n, int a[], char *title );
void i4mat_transpose_print_some ( int m, int n, int a[], int ilo, int jlo, 
  int ihi, int jhi, char *title );
int *i4mat_u1_inverse ( int n, int a[] );
int *i4mat_uniform ( int m, int n, int a, int b, int *seed );
int i4row_compare ( int m, int n, int a[], int i, int j );
void i4row_find_item ( int m, int n, int a[], int item, int *row, int *col );
int *i4row_max ( int m, int n, int a[] );
double *i4row_mean ( int m, int n, int a[] );
int *i4row_min ( int m, int n, int a[] );
void i4row_sort_a ( int m, int n, int a[] );
void i4row_sort_d ( int m, int n, int a[] );
void i4row_sort2_d ( int m, int n, int a[] );
int *i4row_sum ( int m, int n, int a[] );
void i4row_swap ( int m, int n, int a[], int irow1, int irow2 );
double *i4row_variance ( int m, int n, int a[] );
int i4vec_amax ( int n, int a[] );
int i4vec_amax_index ( int n, int a[] );
int i4vec_amin ( int n, int a[] );
int i4vec_amin_index ( int n, int a[] );
int i4vec_aminz ( int n, int a[] );
int i4vec_aminz_index ( int n, int a[] );
int *i4vec_ascend_sub ( int n, int a[], int *length );
bool i4vec_ascends ( int n, int a[] );
void i4vec_axpy ( int n, int ia, int ix[], int incx, int iy[], int incy );
void i4vec_bracket ( int n, int x[], int xval, int *left, int *right );
int i4vec_compare ( int n, int a1[], int a2[] );
void i4vec_copy ( int n, int a1[], int a2[] );
int *i4vec_copy_new ( int n, int a1[] );
int *i4vec_cum ( int n, int a[] );
bool i4vec_descends ( int n, int x[] );
void i4vec_direct_product ( int factor_index, int factor_order, 
  int factor_value[], int factor_num, int point_num, int x[] );
void i4vec_direct_product2 ( int factor_index, int factor_order, 
  int factor_value[], int factor_num, int point_num, int w[] );
int *i4vec_first_index ( int n, int a[] );
int i4vec_frac ( int n, int a[], int k );
int i4vec_gcd ( int n, int a[] );
void i4vec_heap_a ( int n, int a[] );
void i4vec_heap_d ( int n, int a[] );
int i4vec_heap_d_extract ( int *n, int a[] );
void i4vec_heap_d_insert ( int *n, int a[], int value );
int i4vec_heap_d_max ( int n, int a[] );
int *i4vec_histogram ( int n, int a[], int histo_num );
int i4vec_index ( int n, int a[], int aval );
void i4vec_index_delete_all ( int n, int x[], int indx[], int xval, 
  int *n2, int x2[], int indx2[] );
void i4vec_index_delete_dupes ( int n, int x[], int indx[], 
  int *n2, int x2[], int indx2[] );
void i4vec_index_delete_one ( int n, int x[], int indx[], int xval, 
  int *n2, int x2[], int indx2[] );
void i4vec_index_insert ( int *n, int x[], int indx[], int xval );
void i4vec_index_insert_unique ( int *n, int x[], int indx[], int xval );
void i4vec_index_order ( int n, int x[], int indx[] );
void i4vec_index_search ( int n, int x[], int indx[], int xval, int *less, 
  int *equal, int *more );
void i4vec_index_sort_unique ( int n, int x[], int *n2, int x2[], int indx2[] );
int *i4vec_indicator ( int n );
void i4vec_insert ( int n, int a[], int pos, int value );
int i4vec_max ( int n, int *ivec );
int i4vec_max_index ( int n, int a[] );
int i4vec_max_index_last ( int n, int a[] );
double i4vec_mean ( int n, int x[] );
int i4vec_median ( int n, int a[] );
int *i4vec_merge_a ( int na, int a[], int nb, int b[] );
int i4vec_min ( int n, int *ivec );
int i4vec_min_index ( int n, int a[] );
int i4vec_nonzero_count ( int n, int a[] );
void i4vec_nonzero_first ( int n, int x[], int *nz, int indx[] );
int i4vec_order_type ( int n, int x[] );
bool i4vec_pairwise_prime ( int n, int a[] );
void i4vec_part ( int n, int nval, int x[] );
void i4vec_part_quick_a ( int n, int a[], int *l, int *r );
void i4vec_permute ( int n, int p[], int base, int a[] );
void i4vec_permute_uniform ( int n, int a[], int *seed );
void i4vec_print ( int n, int a[], char *title );
void i4vec_print_some ( int n, int a[], int i_lo, int i_hi, char *title );
int i4vec_product ( int n, int a[] );
int i4vec_red ( int n, int a[] );
void i4vec_reverse ( int n, int x[] );
void i4vec_rotate ( int n, int m, int x[] );
int i4vec_run_count ( int n, int a[] );
int i4vec_search_binary_a ( int n, int a[], int b );
int i4vec_search_binary_d ( int n, int a[], int b );
void i4vec_sort_bubble_a ( int n, int a[] );
void i4vec_sort_bubble_d ( int n, int a[] );
void i4vec_sort_heap_a ( int n, int a[] );
void i4vec_sort_heap_d ( int n, int a[] );
int *i4vec_sort_heap_index_a ( int n, int a[] );
int *i4vec_sort_heap_index_d ( int n, int a[] );
void i4vec_sort_insert_a ( int n, int a[] );
void i4vec_sort_insert_d ( int n, int a[] );
void i4vec_sort_quick_a ( int n, int a[] );
void i4vec_sort_shell_a ( int n, int a[] );
void i4vec_sorted_undex ( int x_num, int x_val[], int x_unique_num, int undx[], 
  int xdnu[] );
int i4vec_sorted_unique ( int n, int a[] );
int i4vec_sorted_unique_count ( int n, int a[] );
void i4vec_sorted_unique_hist ( int n, int a[], int maxuniq, int *unique_num, 
  int auniq[], int acount[] );
int i4vec_split ( int n, int a[], int split );
int i4vec_sum ( int n, int a[] );
void i4vec_swap ( int n, int iarray1[], int iarray2[] );
void i4vec_transpose_print ( int n, int a[], char *title );
void i4vec_undex ( int x_num, int x_val[], int x_unique_num, int undx[], 
  int xdnu[] );
int *i4vec_uniform ( int n, int a, int b, int *seed );
int i4vec_unique_count ( int n, int a[] );
int *i4vec_unique_index ( int n, int a[] );
int *i4vec_value_index ( int n, int a[], int value, int max_index, int *n_index );
int i4vec_value_num ( int n, int a[], int value );
double i4vec_variance ( int n, int x[] );
void i4vec_zero ( int n, int a[] );
int *i4vec_zero_new ( int n );
int i4vec2_compare ( int n, int a1[], int a2[], int i, int j );
void i4vec2_print ( int n, int a1[], int a2[], char *title );
void i4vec2_sort_a ( int n, int a1[], int a2[] );
void i4vec2_sort_d ( int n, int a1[], int a2[] );
void i4vec2_sorted_unique ( int n, int a1[], int a2[], int *unique_num );
void i4i4_sort_a ( int i1, int i2, int *j1, int *j2 );
void i4i4i4_sort_a ( int i1, int i2, int i3, int *j1, int *j2, int *j3 );
long long int i8_huge ( );
long long int i8_max ( long long int i1, long long int i2 );
long long int i8_min ( long long int i1, long long int i2 );
long long int i8_power ( long long int i, long long int j );
long long int i8_reverse_bytes ( long long int x );
unsigned long long int i8_xor ( unsigned long long int i, 
  unsigned long long int j );
void ij_next ( int *i, int *j, int n );
void ij_next_gt ( int *i, int *j, int n );
void index_box2_next_2d ( int n1, int n2, int ic, int jc, int *i, int *j, 
  int *more );
void index_box2_next_3d ( int n1, int n2, int n3, int ic, int jc, int kc, 
  int *i, int *j, int *k, bool *more );
int isbn_check ( char *isbn );
void isbn_fill ( char *isbn );
int isbn_to_i4 ( char c );
int iset2_compare ( int x1, int y1, int x2, int y2 );
int lcm_12n ( int n );
void lvec_print ( int n, bool a[], char *title );
int pause_input ( );
void pause_seconds ( int seconds );
bool perm_check ( int n, int p[], int base );
void perm_cycle ( int n, int p[], int *isgn, int *ncycle, int iopt );
int *perm_free ( int npart, int ipart[], int nfree );
void perm_inverse ( int n, int p[] );
void perm_print ( int n, int p[], char *title );
int *perm_uniform ( int n, int base, int *seed );
double pounds_to_kilograms ( double lb );
int prime ( int n );
int prime_ge ( int n );
float r4_abs ( float x );
float r4_atan ( float y, float x );
float r4_epsilon ( );
float r4_exp ( float x );
float r4_huge ( );
float r4_max ( float x, float y );
float r4_min ( float x, float y );
int r4_nint ( float x );
float r4_reverse_bytes ( float x );
void r4_swap ( float *x, float *y );
float r4_tiny ( );
float r4_uniform_01 ( int *seed );
double r8_abs ( double x );
double r8_atan ( double y, double x );
double r8_cas ( double x );
double r8_ceiling ( double x );
double r8_choose ( int n, int k );
double r8_chop ( int place, double x );
complex <double> r8_csqrt ( double x );
double r8_cube_root ( double x );
double r8_diff ( double x, double y, int n );
int r8_digit ( double x, int idigit );
double r8_epsilon ( );
double r8_exp ( double x );
double r8_factorial ( int n );
double r8_factorial2 ( int n );
double r8_floor ( double x );
double r8_fraction ( double x );
double r8_huge ( );
bool r8_in_01 ( double a );
bool r8_is_int ( double r );
double r8_log_10 ( double x );
double r8_log_2 ( double x );
double r8_log_b ( double x, double b );
void r8_mant ( double x, int *s, double *r, int *l );
double r8_max ( double x, double y );
double r8_min ( double x, double y );
double r8_mod ( double x, double y );
double r8_modp ( double x, double y );
double r8_mop ( int i );
int r8_nint ( double x );
double r8_normal_01 ( int *seed );
double r8_pi ( );
double r8_power ( double r, int p );
double r8_power_fast ( double r, int p, int *mults );
double r8_pythag ( double a, double b );
double r8_round2 ( int nplace, double x );
double r8_roundb ( int base, int nplace, double x );
double r8_roundx ( int nplace, double x );
double r8_sign ( double x );
double r8_reverse_bytes ( double x );
void r8_swap ( double *x, double *y );
void r8_swap3 ( double *x, double *y, double *z );
double r8_tiny ( );
double r8_to_r8_discrete ( double r, double rmin, double rmax, int nr );
void r8_to_dhms ( double r, int *d, int *h, int *m, int *s );
int r8_to_i4 ( double x, double xmin, double xmax, int ixmin, int ixmax );
double r8_uniform ( double b, double c, int *seed );
double r8_uniform_01 ( int *seed );
void r8_unswap3 ( double *x, double *y, double *z );
double r8_walsh_1d ( double x, int digit );
double *r82_cheby ( int n, double alo, double ahi );
void r82_print ( double a[2], char *title );
void r82_uniform ( double b, double c, int *seed, double r[] );
void r82poly2_print ( double a, double b, double c, double d, double e, double f );
int r82poly2_type ( double a, double b, double c, double d, double e, double f );
void r82poly2_type_print ( int type );
double *r82vec_max ( int n, double a[] );
double *r82vec_min ( int n, double a[] );
int r82vec_order_type ( int n, double a[] );
void r82vec_part_quick_a ( int n, double a[], int *l, int *r );
void r82vec_permute ( int n, int p[], int base, double a[] );
void r82vec_print ( int n, double a[], char *title );
int *r82vec_sort_heap_index_a ( int n, double a[] );
void r82vec_sort_quick_a ( int n, double a[] );
double *r83vec_max ( int n, double a[] );
double *r83vec_min ( int n, double a[] );
void r83vec_part_quick_a ( int n, double a[], int *l, int *r );
void r83vec_sort_quick_a ( int n, double a[] );
double *r83vec_uniform ( int n, double b, double c, int *seed );
double *r8block_expand_linear ( int l, int m, int n, double x[], int lfat, int mfat, 
  int nfat );
void r8block_print ( int l, int m, int n, double a[], char *title );
int r8col_compare ( int m, int n, double a[], int i, int j );
int r8col_find ( int m, int n, double a[], double x[] );
int *r8col_first_index ( int m, int n, double a[], double tol );
int r8col_insert ( int n_max, int m, int n, double a[], double x[] );
double *r8col_max ( int m, int n, double a[] );
int *r8col_max_index ( int m, int n, double a[] );
double *r8col_mean ( int m, int n, double a[] );
double *r8col_min ( int m, int n, double a[] );
int *r8col_min_index ( int m, int n, double a[] );
void r8col_part_quick_a ( int m, int n, double a[], int *l, int *r );
void r8col_permute ( int m, int n, int p[], int base, double a[] );
void r8col_sort_heap_a ( int m, int n, double a[] );
int *r8col_sort_heap_index_a ( int m, int n, double a[] );
void r8col_sort_quick_a ( int m, int n, double a[] );
void r8col_sorted_undex ( int x_dim, int x_num, double x_val[], 
  int x_unique_num, double tol, int undx[], int xdnu[] );
int r8col_sorted_unique ( int m, int n, double a[] );
int r8col_sorted_unique_count ( int m, int n, double a[], double tol );
void r8col_sortr_a ( int m, int n, double a[], int key );
double *r8col_sum ( int m, int n, double a[] );
void r8col_swap ( int m, int n, double a[], int j1, int j2 );
double *r8col_to_r8vec ( int m, int n, double a[] );
void r8col_undex ( int x_dim, int x_num, double x_val[], int x_unique_num, 
  double tol, int undx[], int xdnu[] );
int r8col_unique_count ( int m, int n, double a[], double tol );
int *r8col_unique_index ( int m, int n, double a[], double tol );
double *r8col_variance ( int m, int n, double a[] );
int r8r8_compare ( double x1, double y1, double x2, double y2 );
void r8r8_print ( double a1, double a2, char *title );
int r8r8r8_compare ( double x1, double y1, double z1, double x2, double y2, 
  double z2 );
void r8r8r8vec_index_insert_unique ( int maxn, int *n, double x[], double y[], 
  double z[], int indx[], double xval, double yval, double zval, int *ival, 
  int *ierror );
void r8r8r8vec_index_search ( int n, double x[], double y[], double z[], 
  int indx[], double xval, double yval, double zval, int *less, int *equal, 
  int *more );
void r8r8vec_index_insert_unique ( int maxn, int *n, double x[], double y[], 
  int indx[], double xval, double yval, int *ival, int *ierror );
void r8r8vec_index_search ( int n, double x[], double y[], int indx[], 
  double xval, double yval, int *less, int *equal, int *more );
double degrees_to_radians ( double angle );
double r8int_to_r8int ( double rmin, double rmax, double r, double r2min, 
  double r2max );
int r8int_to_i4int ( double rmin, double rmax, double r, int imin, int imax );
double *r8mat_cholesky_factor ( int n, double a[] );
double *r8mat_cholesky_solve ( int n, double a[], double b[] );
void r8mat_copy ( int m, int n, double a1[], double a2[] );
double *r8mat_copy_new ( int m, int n, double a1[] );
double r8mat_det ( int n, double a[] );
double r8mat_det_2d ( double a[] );
double r8mat_det_3d ( double a[] );
double r8mat_det_4d ( double a[] );
double r8mat_det_5d ( double a[] );
void r8mat_diag_add_scalar ( int n, double a[], double s );
void r8mat_diag_add_vector ( int n, double a[], double v[] );
double *r8mat_diag_get_vector ( int n, double a[] );
void r8mat_diag_set_scalar ( int n, double a[], double s );
void r8mat_diag_set_vector ( int n, double a[], double v[] );
double *r8mat_expand_linear ( int m, int n, double x[], int mfat, int nfat );
double *r8mat_expand_linear2 ( int m, int n, double a[], int m2, int n2 );
double *r8mat_givens_post ( int n, double a[], int row, int col );
double *r8mat_givens_pre ( int n, double a[], int row, int col );
double *r8mat_hess ( double (*fx )( int n, double x[] ), int n, double x[] );
void r8mat_house_axh ( int n, double a[], double v[] );
double *r8mat_house_axh_new ( int n, double a[], double v[] );
double *r8mat_house_form ( int n, double v[] );
double *r8mat_house_hxa ( int n, double a[], double v[] );
double *r8mat_house_post ( int n, double a[], int row, int col );
double *r8mat_house_pre ( int n, double a[], int row, int col );
double *r8mat_identity ( int n );
bool r8mat_in_01 ( int m, int n, double a[] );
double *r8mat_indicator ( int m, int n );
double *r8mat_inverse_2d ( double a[] );
double *r8mat_inverse_3d ( double a[] );
double *r8mat_inverse_4d ( double a[] );
double *r8mat_jac ( int m, int n, double eps, 
  double *(*fx) ( int m, int n, double x[] ), double x[] );
double *r8mat_l_inverse ( int n, double a[] );
void r8mat_l_print ( int m, int n, double a[], char *title );
double *r8mat_l_solve ( int n, double a[], double b[] );
double *r8mat_l1_inverse ( int n, double a[] );
double *r8mat_lt_solve ( int n, double a[], double b[] );
void r8mat_lu ( int m, int n, double a[], double l[], double p[], double u[] );
double r8mat_max ( int m, int n, double a[] );
void r8mat_max_index ( int m, int n, double a[], int *i_max, int *j_max );
double r8mat_maxcol_minrow ( int m, int n, double a[] );
double r8mat_maxrow_mincol ( int m, int n, double a[] );
double r8mat_min ( int m, int n, double a[] );
void r8mat_min_index ( int m, int n, double a[], int *i_min, int *j_min );
double r8mat_mincol_maxrow ( int m, int n, double a[] );
double r8mat_minrow_maxcol ( int m, int n, double a[] );
double *r8mat_mm ( int n1, int n2, int n3, double a[], double b[] );
double *r8mat_mtv ( int m, int n, double a[], double x[] );
void r8mat_mtxv ( int m, int n, double a[], double x[], double atx[] );
double *r8mat_mv ( int m, int n, double a[], double x[] );
void r8mat_mxm ( int n1, int n2, int n3, double a[], double b[], double c[] );
double *r8mat_mxm_new ( int n1, int n2, int n3, double a[], double b[] );
void r8mat_mxv ( int m, int n, double a[], double x[], double ax[] );
void r8mat_nint ( int m, int n, double a[] );
double r8mat_norm_eis ( int m, int n, double a[] );
double r8mat_norm_fro ( int m, int n, double a[] );
double *r8mat_nullspace ( int m, int n, double a[], int nullspace_size );
int r8mat_nullspace_size ( int m, int n, double a[] );
double *r8mat_orth_uniform ( int n, int *seed );
void r8mat_plot ( int m, int n, double a[], char *title );
char r8mat_plot_symbol ( double r );
double *r8mat_poly_char ( int n, double a[] );
double *r8mat_power ( int n, double a[], int npow );
void r8mat_power_method ( int n, double a[], double *r, double v[] );
void r8mat_print ( int m, int n, double a[], char *title );
void r8mat_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi, 
  int jhi, char *title );
void r8mat_ref ( int m, int n, double a[] );
void r8mat_rref ( int m, int n, double a[] );
int r8mat_solve ( int n, int nrhs, double a[] );
double *r8mat_solve_2d ( double a[], double b[], double *det );
double *r8mat_solve_3d ( double a[], double b[], double *det );
double *r8mat_solve2 ( int n, double a[], double b[], int *ierror );
double *r8mat_symm_eigen ( int n, double x[], double q[] );
void r8mat_symm_jacobi ( int n, double a[] );
int r8mat_to_r8plu ( int n, double a[], int pivot[], double lu[] );
double r8mat_trace ( int n, double a[] );
double *r8mat_transpose ( int m, int n, double a[] );
void r8mat_transpose_in_place ( int n, double a[] );
void r8mat_transpose_print ( int m, int n, double a[], char *title );
void r8mat_transpose_print_some ( int m, int n, double a[], int ilo, int jlo, 
  int ihi, int jhi, char *title );
double *r8mat_u_inverse ( int n, double a[] );
double *r8mat_u1_inverse ( int n, double a[] );
double *r8mat_uniform ( int m, int n, double b, double c, int *seed );
double *r8mat_uniform_01 ( int m, int n, int *seed );
double *r8mat_vand2 ( int n, double x[] );
void r8mat_zero ( int m, int n, double a[] );
double *r8mat_zero_new ( int m, int n );
double r8plu_det ( int n, int pivot[], double lu[] );
void r8plu_inverse ( int n, int pivot[], double lu[], double a_inverse[] );
void r8plu_mul ( int n, int pivot[], double lu[], double x[], double b[] );
void r8plu_sol ( int n, int pivot[], double lu[], double b[], double x[] );
void r8plu_to_r8mat ( int n, int pivot[], double lu[], double a[] );
int r8poly_degree ( int na, double a[] );
double *r8poly_deriv ( int n, double c[], int p );
double r8poly_lagrange_0 ( int npol, double xpol[], double xval );
double r8poly_lagrange_1 ( int npol, double xpol[], double xval );
double r8poly_lagrange_2 ( int npol, double xpol[], double xval );
double *r8poly_lagrange_coef ( int npol, int ipol, double xpol[] );
void r8poly_lagrange_factor ( int npol, double xpol[], double xval, 
  double *wval, double *dwdx );
int r8poly_lagrange_val ( int npol, int ipol, double xpol[], double xval, 
  double *pval, double *dpdx );
int r8poly_order ( int na, double a[] );
void r8poly_print ( int n, double a[], char *title );
void r8poly_shift ( double scale, double shift, int n, double poly_cof[] );
double r8poly_val_horner ( int n, double c[], double x );
int r8poly2_ex ( double x1, double y1, double x2, double y2, double x3, double y3, 
  double *x, double *y );
int r8poly2_ex2 ( double x1, double y1, double x2, double y2, double x3, double y3, 
  double *x, double *y, double *a, double *b, double *c );
void r8poly2_root ( double a, double b, double c, complex <double> *r1, 
  complex <double> *r2 );
void r8poly2_rroot ( double a, double b, double c, double *r1, double *r2 );
void r8poly2_val ( double x1, double y1, double x2, double y2, 
  double x3, double y3, double x, double *y, double *yp, double *ypp );
void r8poly2_val2 ( int ndata, double tdata[], 
  double ydata[], int left, double tval, double *yval );
void r8poly3_root ( double a, double b, double c, double d, complex <double> *r1, 
  complex <double> *r2, complex <double> *r3 );
void r8poly4_root ( double a, double b, double c, double d, double e, 
  complex <double> *r1, complex <double> *r2, complex <double> *r3, 
  complex <double> *r4 );
double *r8row_max ( int m, int n, double a[] );
double *r8row_mean ( int m, int n, double a[] );
double *r8row_min ( int m, int n, double a[] );
double *r8row_sum ( int m, int n, double a[] );
void r8row_swap ( int m, int n, double a[], int irow1, int irow2 );
double *r8row_to_r8vec ( int m, int n, double a[] );
double *r8row_variance ( int m, int n, double a[] );
void r8slmat_print ( int m, int n, double a[], char *title );
void r8vec_01_to_ab ( int n, double a[], double amax, double amin );
void r8vec_ab_to_01 ( int n, double a[] );
double *r8vec_ab_to_cd ( int n, double a[], double bmin, double bmax );
double r8vec_amax ( int n, double r8vec[] );
int r8vec_amax_index ( int n, double a[] );
double r8vec_amin ( int n, double r8vec[] );
int r8vec_amin_index ( int n, double a[] );
void r8vec_bracket ( int n, double x[], double xval, int *left, 
  int *right );
void r8vec_bracket2 ( int n, double x[], double xval, int start, int *left, 
  int *right );
void r8vec_bracket3 ( int n, double t[], double tval, int *left );
double r8vec_circular_variance ( int n, double x[] );
int r8vec_compare ( int n, double a[], double b[] );
double *r8vec_convolve_circ ( int n, double x[], double y[] );
void r8vec_copy ( int n, double a1[], double a2[] );
double *r8vec_copy_new ( int n, double a1[] );
double *r8vec_cross_3d ( double v1[3], double v2[3] );
double *r8vec_dif ( int n, double h );
void r8vec_direct_product ( int factor_index, int factor_order, 
  double factor_value[], int factor_num, int point_num, double x[] );
void r8vec_direct_product2 ( int factor_index, int factor_order, 
  double factor_value[], int factor_num, int point_num, double w[] );
double r8vec_distance ( int dim_num, double v1[], double v2[] );
bool r8vec_distinct ( int n, double x[] );
void r8vec_divide ( int n, double a[], double s );
double r8vec_dot ( int n, double a1[], double a2[] );
bool r8vec_eq ( int n, double a1[], double a2[] );
double *r8vec_even ( int n, double alo, double ahi );
double r8vec_even_select ( int n, double xlo, double xhi, int ival );
void r8vec_even2 ( int maxval, int nfill[], int nold, double xold[], 
  int *nval, double xval[] );
void r8vec_even3 ( int nold, int nval, double xold[], double xval[] );
double *r8vec_expand_linear ( int n, double x[], int fat );
int *r8vec_first_index ( int n, double a[], double tol );
double r8vec_frac ( int n, double a[], int k );
double *r8vec_fraction ( int n, double x[] );
bool r8vec_gt ( int n, double a1[], double a2[] );
void r8vec_heap_a ( int n, double a[] );
void r8vec_heap_d ( int n, double a[] );
int *r8vec_histogram ( int n, double a[], double a_lo, double a_hi, int histo_num );
double *r8vec_house_column ( int n, double a[], int k );
bool r8vec_in_01 ( int n, double a[] );
void r8vec_index_delete_all ( int n, double x[], int indx[], double xval, 
  int *n2, double x2[], int indx2[] );
void r8vec_index_delete_dupes ( int n, double x[], int indx[], 
  int *n2, double x2[], int indx2[] );
void r8vec_index_delete_one ( int n, double x[], int indx[], double xval, 
  int *n2, double x2[], int indx2[] );
void r8vec_index_insert ( int *n, double x[], int indx[], double xval );
void r8vec_index_insert_unique ( int *n, double x[], int indx[], double xval );
void r8vec_index_order ( int n, double x[], int indx[] );
void r8vec_index_search ( int n, double x[], int indx[], double xval, int *less, 
  int *equal, int *more );
void r8vec_index_sort_unique ( int n, double x[], int *n2, double x2[], int indx2[] );
double *r8vec_indicator ( int n );
void r8vec_insert ( int n, double a[], int pos, double value );
bool r8vec_is_int ( int n, double a[] );
double r8vec_length ( int dim_num, double x[] );
bool r8vec_lt ( int n, double a1[], double a2[] );
void r8vec_mask_print ( int n, double a[], int mask_num, int mask[], char *title );
double r8vec_max ( int n, double r8vec[] );
int r8vec_max_index ( int n, double a[] );
double r8vec_mean ( int n, double x[] );
double r8vec_median ( int n, double a[] );
double r8vec_min ( int n, double r8vec[] );
int r8vec_min_index ( int n, double a[] );
bool r8vec_mirror_next ( int n, double a[] );
double *r8vec_nint ( int n, double a[] );
double r8vec_norm_l1 ( int n, double a[] );
double r8vec_norm_l2 ( int n, double a[] );
double r8vec_norm_li ( int n, double a[] );
double r8vec_norm_lp ( int n, double a[], double p );
double *r8vec_normal_01 ( int n, int *seed );
int r8vec_order_type ( int n, double x[] );
void r8vec_part_quick_a ( int n, double a[], int *l, int *r );
void r8vec_permute ( int n, int p[], int base, double a[] );
void r8vec_permute_uniform ( int n, double a[], int *seed );
void r8vec_polarize ( int n, double a[], double p[], double a_normal[], 
  double a_parallel[] );
void r8vec_print ( int n, double a[], char *title );
void r8vec_print_some ( int n, double a[], int i_lo, int i_hi, char *title );
double r8vec_product ( int n, double a[] );
void r8vec_range ( int n, double x[], double xmin, double xmax, double y[], 
  double *ymin, double *ymax );
void r8vec_range_2 ( int n, double a[], double *amin, double *amax );
double *r8vec_read ( char *input_file, int n );
int r8vec_read_size ( char *input_file );
void r8vec_reverse ( int n, double a[] );
void r8vec_rotate ( int n, double a[], int m );
int r8vec_search_binary_a ( int n, double a[], double aval );
void r8vec_sort_bubble_a ( int n, double a[] );
void r8vec_sort_bubble_d ( int n, double a[] );
void r8vec_sort_heap_a ( int n, double a[] );
void r8vec_sort_heap_d ( int n, double a[] );
int *r8vec_sort_heap_index_a ( int n, double a[] );
int *r8vec_sort_heap_index_d ( int n, double a[] );
int *r8vec_sort_heap_mask_a ( int n, double a[], int mask_num, int mask[] );
void r8vec_sort_insert_a ( int n, double a[] );
int *r8vec_sort_insert_index_a ( int n, double a[] );
void r8vec_sort_quick_a ( int n, double a[] );
void r8vec_sort_shell_a ( int n, double a[] );
double *r8vec_sorted_merge_a ( int na, double a[], int nb, double b[], int *nc );
int r8vec_sorted_nearest ( int n, double a[], double value );
void r8vec_sorted_split ( int n, double a[], double split, int *i_lt, int *i_gt );
void r8vec_sorted_undex ( int x_num, double x_val[], int x_unique_num, 
  double tol, int undx[], int xdnu[] );
double *r8vec_sorted_unique ( int n, double a[], double tol, int *unique_num );
int r8vec_sorted_unique_count ( int n, double a[], double tol );
void r8vec_sorted_unique_hist ( int n, double a[], double tol, int maxuniq, 
  int *unique_num, double auniq[], int acount[] );
int r8vec_split ( int n, double a[], double split );
double r8vec_std ( int n, double a[] );
double r8vec_sum ( int n, double a[] );
void r8vec_swap ( int n, double a1[], double a2[] );
void r8vec_undex ( int x_num, double x_val[], int x_unique_num, double tol,
  int undx[], int xdnu[] );
double *r8vec_uniform ( int n, double b, double c, int *seed );
double *r8vec_uniform_01 ( int n, int *seed );
int r8vec_unique_count ( int n, double a[], double tol );
int *r8vec_unique_index ( int n, double a[], double tol );
void r8vec_unit_sum ( int n, double a[] );
double r8vec_variance ( int n, double x[] );
void r8vec_write ( int n, double r[], char *output_file );
void r8vec_zero ( int n, double a1[] );
double *r8vec_zero_new ( int n );
int r8vec2_compare ( int n, double a1[], double a2[], int i, int j );
void r8vec2_print ( int n, double a1[], double a2[], char *title );
void r8vec2_print_some ( int n, double x1[], double x2[], int max_print, 
  char *title );
void r8vec2_sort_a ( int n, double a1[], double a2[] );
void r8vec2_sort_d ( int n, double a1[], double a2[] );
int *r8vec2_sort_heap_index_a ( int n, double x[], double y[] );
void r8vec2_sorted_unique ( int n, double a1[], double a2[], int *unique_num );
void r8vec2_sorted_unique_index ( int n, double a1[], double a2[], 
  int *unique_num, int indx[] );
int r8vec2_sum_max_index ( int n, double a[], double b[] );
void r8vec3_print ( int n, double a1[], double a2[], double a3[], char *title );
double *r8vec3_vtriple ( double v1[3], double v2[3], double v3[3] );
double radians_to_degrees ( double angle );
unsigned long rand_initialize ( unsigned long seed );
unsigned long random_initialize ( unsigned long seed );
void rat_factor ( int m, int n, int factor_max, int *factor_num, int factor[], 
  int power[], int *mleft, int *nleft );
double rickey ( int ab, int bb, int er, double f, int h, int hb, int hp, 
  int r, int so, int tb );
int *roots_to_i4poly ( int n, int x[] );
double *roots_to_r8poly ( int n, double x[] );
bool s_eqi ( char *s1, char *s2 );
int s_len_trim ( char* s );
int s_to_i4 ( char *s, int *last, bool *error );
bool s_to_i4vec ( char *s, int n, int ivec[] );
double s_to_r8 ( char *s, int *lchar, bool *error );
bool s_to_r8vec ( char *s, int n, double rvec[] );
void sort_heap_external ( int n, int *indx, int *i, int *j, int isgn );
void timestamp ( );
void tuple_next2 ( int n, int xmin[], int xmax[], int x[], int *rank );
double *tvec_even ( int nt );
double *tvec_even2 ( int nt );
double *tvec_even3 ( int nt );
double *tvec_even_bracket ( int nt, double theta1, double theta2 );
double *tvec_even_bracket2 ( int nt, double theta1, double theta2 );
double *tvec_even_bracket3 ( int nt, double theta1, double theta2 );
int upc_check_digit ( int p, int l, int r );