#+TITLE: LIZFCM Software Manual (v0.5)
#+AUTHOR: Elizabeth Hunt
#+LATEX_HEADER: \notindent \notag  \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+STARTUP: entitiespretty fold inlineimages

* Design
The LIZFCM static library (at [[https://github.com/Simponic/math-4610]]) is a successor to my
attempt at writing codes for the Fundamentals of Computational Mathematics course in Common
Lisp, but the effort required to meet the requirement of creating a static library became
too difficult to integrate outside of the ~ASDF~ solution that Common Lisp already brings
to the table.

All of the work established in ~deprecated-cl~ has been painstakingly translated into
the C programming language. I have a couple tenets for its design:

+ Implementations of routines should all be done immutably in respect to arguments.
+ Functional programming is good (it's... rough in C though).
+ Routines are separated into "modules" that follow a form of separation of concerns
  in files, and not individual files per function.

* Compilation
A provided ~Makefile~ is added for convencience. It has been tested on an ~arm~-based M1 machine running
MacOS as well as ~x86~ Arch Linux.

1. ~cd~ into the root of the repo
2. ~make~
   
Then, as of homework 5, the testing routines are provided in ~test~ and utilize the
~utest~ "micro"library. They compile to a binary in ~./dist/lizfcm.test~.

Execution of the Makefile will perform compilation of individual routines.

But, in the requirement of manual intervention (should the little alien workers
inside the computer fail to do their job), one can use the following command to
produce an object file:

\begin{verbatim}
  gcc -Iinc/ -lm -Wall -c src/<the_routine>.c -o build/<the_routine>.o
\end{verbatim}

Which is then bundled into a static library in ~lib/lizfcm.a~ and can be linked
in the standard method.

* The LIZFCM API
** Simple Routines
*** ~smaceps~
+ Author: Elizabeth Hunt
+ Name: ~smaceps~
+ Location: ~src/maceps.c~
+ Input: none
+ Output: a ~float~ returning the specific "Machine Epsilon" of a machine on a
  single precision floating point number at which it becomes "indistinguishable".

#+BEGIN_SRC c
float smaceps() {
  float one = 1.0;
  float machine_epsilon = 1.0;
  float one_approx = one + machine_epsilon;

  while (fabsf(one_approx - one) > 0) {
    machine_epsilon /= 2;
    one_approx = one + machine_epsilon;
  }

  return machine_epsilon;
}
#+END_SRC

*** ~dmaceps~
+ Author: Elizabeth Hunt
+ Name: ~dmaceps~
+ Location: ~src/maceps.c~
+ Input: none
+ Output: a ~double~ returning the specific "Machine Epsilon" of a machine on a
  double precision floating point number at which it becomes "indistinguishable".

#+BEGIN_SRC c
double dmaceps() {
  double one = 1.0;
  double machine_epsilon = 1.0;
  double one_approx = one + machine_epsilon;

  while (fabs(one_approx - one) > 0) {
    machine_epsilon /= 2;
    one_approx = one + machine_epsilon;
  }

  return machine_epsilon;
}
#+END_SRC

** Derivative Routines
*** ~central_derivative_at~
+ Author: Elizabeth Hunt
+ Name: ~central_derivative_at~
+ Location: ~src/approx_derivative.c~
+ Input:
  - ~f~ is a pointer to a one-ary function that takes a double as input and produces
    a double as output
  - ~a~ is the domain value at which we approximate ~f'~
  - ~h~ is the step size
+ Output: a ~double~ of the approximate value of ~f'(a)~ via the central difference
  method.

#+BEGIN_SRC c
double central_derivative_at(double (*f)(double), double a, double h) {
  assert(h > 0);

  double x2 = a + h;
  double x1 = a - h;

  double y2 = f(x2);
  double y1 = f(x1);

  return (y2 - y1) / (x2 - x1);
}
#+END_SRC

*** ~forward_derivative_at~
+ Author: Elizabeth Hunt
+ Name: ~forward_derivative_at~
+ Location: ~src/approx_derivative.c~
+ Input:
  - ~f~ is a pointer to a one-ary function that takes a double as input and produces
    a double as output
  - ~a~ is the domain value at which we approximate ~f'~
  - ~h~ is the step size
+ Output: a ~double~ of the approximate value of ~f'(a)~ via the forward difference
  method.

#+BEGIN_SRC c
double forward_derivative_at(double (*f)(double), double a, double h) {
  assert(h > 0);

  double x2 = a + h;
  double x1 = a;

  double y2 = f(x2);
  double y1 = f(x1);

  return (y2 - y1) / (x2 - x1);
}
#+END_SRC

*** ~backward_derivative_at~
+ Author: Elizabeth Hunt
+ Name: ~backward_derivative_at~
+ Location: ~src/approx_derivative.c~
+ Input:
  - ~f~ is a pointer to a one-ary function that takes a double as input and produces
    a double as output
  - ~a~ is the domain value at which we approximate ~f'~
  - ~h~ is the step size
+ Output: a ~double~ of the approximate value of ~f'(a)~ via the backward difference
  method.

#+BEGIN_SRC c
double backward_derivative_at(double (*f)(double), double a, double h) {
  assert(h > 0);

  double x2 = a;
  double x1 = a - h;

  double y2 = f(x2);
  double y1 = f(x1);

  return (y2 - y1) / (x2 - x1);
}
#+END_SRC

** Vector Routines
*** Vector Arithmetic: ~add_v, minus_v~
+ Author: Elizabeth Hunt
+ Name(s): ~add_v~, ~minus_v~
+ Location: ~src/vector.c~
+ Input: two pointers to locations in memory wherein ~Array_double~'s lie
+ Output: a pointer to a new ~Array_double~ as the result of addition or subtraction
  of the two input ~Array_double~'s

#+BEGIN_SRC c
Array_double *add_v(Array_double *v1, Array_double *v2) {
  assert(v1->size == v2->size);

  Array_double *sum = copy_vector(v1);
  for (size_t i = 0; i < v1->size; i++)
    sum->data[i] += v2->data[i];
  return sum;
}

Array_double *minus_v(Array_double *v1, Array_double *v2) {
  assert(v1->size == v2->size);

  Array_double *sub = InitArrayWithSize(double, v1->size, 0);
  for (size_t i = 0; i < v1->size; i++)
    sub->data[i] = v1->data[i] - v2->data[i];
  return sub;
}
#+END_SRC

*** Norms: ~l1_norm~, ~l2_norm~, ~linf_norm~
+ Author: Elizabeth Hunt
+ Name(s): ~l1_norm~, ~l2_norm~, ~linf_norm~
+ Location: ~src/vector.c~
+ Input: a pointer to a location in memory wherein an ~Array_double~ lies
+ Output: a ~double~ representing the value of the norm the function applies

#+BEGIN_SRC c
double l1_norm(Array_double *v) {
  double sum = 0;
  for (size_t i = 0; i < v->size; ++i)
    sum += fabs(v->data[i]);
  return sum;
}

double l2_norm(Array_double *v) {
  double norm = 0;
  for (size_t i = 0; i < v->size; ++i)
    norm += v->data[i] * v->data[i];
  return sqrt(norm);
}

double linf_norm(Array_double *v) {
  assert(v->size > 0);
  double max = v->data[0];
  for (size_t i = 0; i < v->size; ++i)
    max = c_max(v->data[i], max);
  return max;
}
#+END_SRC

*** ~vector_distance~
+ Author: Elizabeth Hunt
+ Name: ~vector_distance~
+ Location: ~src/vector.c~
+ Input: two pointers to locations in memory wherein ~Array_double~'s lie, and a pointer to a
  one-ary function ~norm~ taking as input a pointer to an ~Array_double~ and returning a double
  representing the norm of that ~Array_double~

#+BEGIN_SRC c
double vector_distance(Array_double *v1, Array_double *v2,
                       double (*norm)(Array_double *)) {
  Array_double *minus = minus_v(v1, v2);
  double dist = (*norm)(minus);
  free(minus);
  return dist;
}
#+END_SRC

*** Distances: ~l1_distance~, ~l2_distance~, ~linf_distance~
+ Author: Elizabeth Hunt
+ Name(s): ~l1_distance~, ~l2_distance~, ~linf_distance~
+ Location: ~src/vector.c~
+ Input: two pointers to locations in memory wherein ~Array_double~'s lie, and the distance
  via the corresponding ~l1~, ~l2~, or ~linf~ norms
+ Output: A ~double~ representing the distance between the two ~Array_doubles~'s by the given
  norm.
  
#+BEGIN_SRC c
double l1_distance(Array_double *v1, Array_double *v2) {
  return vector_distance(v1, v2, &l1_norm);
}

double l2_distance(Array_double *v1, Array_double *v2) {
  return vector_distance(v1, v2, &l2_norm);
}

double linf_distance(Array_double *v1, Array_double *v2) {
  return vector_distance(v1, v2, &linf_norm);
}
#+END_SRC

*** ~sum_v~
+ Author: Elizabeth Hunt
+ Name: ~sum_v~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~
+ Output: a ~double~ representing the sum of all the elements of an ~Array_double~

#+BEGIN_SRC c
double sum_v(Array_double *v) {
  double sum = 0;
  for (size_t i = 0; i < v->size; i++)
    sum += v->data[i];
  return sum;
}
#+END_SRC

*** ~scale_v~
+ Author: Elizabeth Hunt
+ Name: ~scale_v~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~ and a scalar ~double~ to scale the vector
+ Output: a pointer to a new ~Array_double~ of the scaled input ~Array_double~

#+BEGIN_SRC c
Array_double *scale_v(Array_double *v, double m) {
  Array_double *copy = copy_vector(v);
  for (size_t i = 0; i < v->size; i++)
    copy->data[i] *= m;
  return copy;
}
#+END_SRC

*** ~free_vector~
+ Author: Elizabeth Hunt
+ Name: ~free_vector~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~
+ Output: nothing.
+ Side effect: free the memory of the reserved ~Array_double~ on the heap

#+BEGIN_SRC c
void free_vector(Array_double *v) {
  free(v->data);
  free(v);
}
#+END_SRC

*** ~add_element~
+ Author: Elizabeth Hunt
+ Name: ~add_element~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~
+ Output: a new ~Array_double~ with element ~x~ appended.

#+BEGIN_SRC c
Array_double *add_element(Array_double *v, double x) {
  Array_double *pushed = InitArrayWithSize(double, v->size + 1, 0.0);
  for (size_t i = 0; i < v->size; ++i)
    pushed->data[i] = v->data[i];
  pushed->data[v->size] = x;
  return pushed;
}
#+END_SRC

*** ~slice_element~
+ Author: Elizabeth Hunt
+ Name: ~slice_element~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~
+ Output: a new ~Array_double~ with element ~x~ sliced.

#+BEGIN_SRC c
Array_double *slice_element(Array_double *v, size_t x) {
  Array_double *sliced = InitArrayWithSize(double, v->size - 1, 0.0);
  for (size_t i = 0; i < v->size - 1; ++i)
    sliced->data[i] = i >= x ? v->data[i + 1] : v->data[i];
  return sliced;
}
#+END_SRC

*** ~copy_vector~
+ Author: Elizabeth Hunt
+ Name: ~copy_vector~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~
+ Output: a pointer to a new ~Array_double~ whose ~data~ and ~size~ are copied from the input
  ~Array_double~

#+BEGIN_SRC c
Array_double *copy_vector(Array_double *v) {
  Array_double *copy = InitArrayWithSize(double, v->size, 0.0);
  for (size_t i = 0; i < copy->size; ++i)
    copy->data[i] = v->data[i];
  return copy;
}
#+END_SRC

*** ~format_vector_into~
+ Author: Elizabeth Hunt
+ Name: ~format_vector_into~
+ Location: ~src/vector.c~
+ Input: a pointer to an ~Array_double~ and a pointer to a c-string ~s~ to "print" the vector out
  into
+ Output: nothing.
+ Side effect: overwritten memory into ~s~

#+BEGIN_SRC c
void format_vector_into(Array_double *v, char *s) {
  if (v->size == 0) {
    strcat(s, "empty");
    return;
  }

  for (size_t i = 0; i < v->size; ++i) {
    char num[64];
    strcpy(num, "");

    sprintf(num, "%f,", v->data[i]);
    strcat(s, num);
  }
  strcat(s, "\n");
}
#+END_SRC

** Matrix Routines
*** ~lu_decomp~
+ Author: Elizabeth Hunt
+ Name: ~lu_decomp~
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ $m$ to decompose into a lower triangular and upper triangular
  matrix $L$, $U$, respectively such that $LU = m$.
+ Output: a pointer to the location in memory in which two ~Matrix_double~'s reside: the first
  representing $L$, the second, $U$.
+ Errors: Fails assertions when encountering a matrix that cannot be
  decomposed

#+BEGIN_SRC c
Matrix_double **lu_decomp(Matrix_double *m) {
  assert(m->cols == m->rows);

  Matrix_double *u = copy_matrix(m);
  Matrix_double *l_empt = InitMatrixWithSize(double, m->rows, m->cols, 0.0);
  Matrix_double *l = put_identity_diagonal(l_empt);
  free_matrix(l_empt);

  Matrix_double **u_l = malloc(sizeof(Matrix_double *) * 2);

  for (size_t y = 0; y < m->rows; y++) {
    if (u->data[y]->data[y] == 0) {
      printf("ERROR: a pivot is zero in given matrix\n");
      assert(false);
    }
  }

  if (u && l) {
    for (size_t x = 0; x < m->cols; x++) {
      for (size_t y = x + 1; y < m->rows; y++) {
        double denom = u->data[x]->data[x];

        if (denom == 0) {
          printf("ERROR: non-factorable matrix\n");
          assert(false);
        }

        double factor = -(u->data[y]->data[x] / denom);

        Array_double *scaled = scale_v(u->data[x], factor);
        Array_double *added = add_v(scaled, u->data[y]);
        free_vector(scaled);
        free_vector(u->data[y]);

        u->data[y] = added;
        l->data[y]->data[x] = -factor;
      }
    }
  }

  u_l[0] = u;
  u_l[1] = l;
  return u_l;
}
#+END_SRC
*** ~bsubst~
+ Author: Elizabeth Hunt
+ Name: ~bsubst~
+ Location: ~src/matrix.c~
+ Input: a pointer to an upper-triangular ~Matrix_double~ $u$ and a ~Array_double~
  $b$
+ Output: a pointer to a new ~Array_double~ whose entries are given by performing
  back substitution

#+BEGIN_SRC c
Array_double *bsubst(Matrix_double *u, Array_double *b) {
  assert(u->rows == b->size && u->cols == u->rows);

  Array_double *x = copy_vector(b);
  for (int64_t row = b->size - 1; row >= 0; row--) {
    for (size_t col = b->size - 1; col > row; col--)
      x->data[row] -= x->data[col] * u->data[row]->data[col];
    x->data[row] /= u->data[row]->data[row];
  }
  return x;
}
#+END_SRC
*** ~fsubst~
+ Author: Elizabeth Hunt
+ Name: ~fsubst~
+ Location: ~src/matrix.c~
+ Input: a pointer to a lower-triangular ~Matrix_double~ $l$ and a ~Array_double~
  $b$
+ Output: a pointer to a new ~Array_double~ whose entries are given by performing
  forward substitution

#+BEGIN_SRC c
Array_double *fsubst(Matrix_double *l, Array_double *b) {
  assert(l->rows == b->size && l->cols == l->rows);

  Array_double *x = copy_vector(b);

  for (size_t row = 0; row < b->size; row++) {
    for (size_t col = 0; col < row; col++)
      x->data[row] -= x->data[col] * l->data[row]->data[col];
    x->data[row] /= l->data[row]->data[row];
  }

  return x;
}
#+END_SRC

*** ~solve_matrix_lu_bsubst~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ $m$ and a pointer to an ~Array_double~ $b$
+ Output: $x$ such that $mx = b$ if such a solution exists (else it's non LU-factorable as discussed
  above)

Here we make use of forward substitution to first solve $Ly = b$ given $L$ as the $L$ factor in
~lu_decomp~. Then we use back substitution to solve $Ux = y$ for $x$ similarly given $U$.

Then, $LUx = b$, thus $x$ is a solution.

#+BEGIN_SRC c
Array_double *solve_matrix_lu_bsubst(Matrix_double *m, Array_double *b) {
  assert(b->size == m->rows);
  assert(m->rows == m->cols);

  Array_double *x = copy_vector(b);
  Matrix_double **u_l = lu_decomp(m);
  Matrix_double *u = u_l[0];
  Matrix_double *l = u_l[1];

  Array_double *b_fsub = fsubst(l, b);
  x = bsubst(u, b_fsub);
  free_vector(b_fsub);

  free_matrix(u);
  free_matrix(l);
  free(u_l);

  return x;
}
#+END_SRC

*** ~gaussian_elimination~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ $m$ 
+ Output: a pointer to a copy of $m$ in reduced echelon form

This works by finding the row with a maximum value in the column $k$. Then, it uses that as a pivot, and
applying reduction to all other rows. The general idea is available at [[https://en.wikipedia.org/wiki/Gaussian_elimination]].

#+BEGIN_SRC c
Matrix_double *gaussian_elimination(Matrix_double *m) {
  uint64_t h = 0;
  uint64_t k = 0;

  Matrix_double *m_cp = copy_matrix(m);

  while (h < m_cp->rows && k < m_cp->cols) {
    uint64_t max_row = 0;
    double total_max = 0.0;

    for (uint64_t row = h; row < m_cp->rows; row++) {
      double this_max = c_max(fabs(m_cp->data[row]->data[k]), total_max);
      if (c_max(this_max, total_max) == this_max) {
        max_row = row;
      }
    }

    if (max_row == 0) {
      k++;
      continue;
    }

    Array_double *swp = m_cp->data[max_row];
    m_cp->data[max_row] = m_cp->data[h];
    m_cp->data[h] = swp;

    for (uint64_t row = h + 1; row < m_cp->rows; row++) {
      double factor = m_cp->data[row]->data[k] / m_cp->data[h]->data[k];
      m_cp->data[row]->data[k] = 0.0;

      for (uint64_t col = k + 1; col < m_cp->cols; col++) {
        m_cp->data[row]->data[col] -= m_cp->data[h]->data[col] * factor;
      }
    }

    h++;
    k++;
  }

  return m_cp;
}
#+END_SRC

*** ~solve_matrix_gaussian~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ $m$ and a target ~Array_double~ $b$
+ Output: a pointer to a vector $x$ being the solution to the equation $mx = b$

We first perform ~gaussian_elimination~ after augmenting $m$ and $b$. Then, as $m$ is in reduced echelon form, it's an upper
triangular matrix, so we can perform back substitution to compute $x$.

#+BEGIN_SRC c
Array_double *solve_matrix_gaussian(Matrix_double *m, Array_double *b) {
  assert(b->size == m->rows);
  assert(m->rows == m->cols);

  Matrix_double *m_augment_b = add_column(m, b);
  Matrix_double *eliminated = gaussian_elimination(m_augment_b);

  Array_double *b_gauss = col_v(eliminated, m->cols);
  Matrix_double *u = slice_column(eliminated, m->rows);

  Array_double *solution = bsubst(u, b_gauss);

  free_matrix(m_augment_b);
  free_matrix(eliminated);
  free_matrix(u);
  free_vector(b_gauss);

  return solution;
}
#+END_SRC


*** ~m_dot_v~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ $m$ and ~Array_double~ $v$
+ Output: the dot product $mv$ as an ~Array_double~
  
#+BEGIN_SRC c
Array_double *m_dot_v(Matrix_double *m, Array_double *v) {
  assert(v->size == m->cols);

  Array_double *product = copy_vector(v);

  for (size_t row = 0; row < v->size; ++row)
    product->data[row] = v_dot_v(m->data[row], v);

  return product;
}
#+END_SRC

*** ~put_identity_diagonal~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~
+ Output: a pointer to a copy to ~Matrix_double~ whose diagonal is full of 1's

#+BEGIN_SRC c
Matrix_double *put_identity_diagonal(Matrix_double *m) {
  assert(m->rows == m->cols);
  Matrix_double *copy = copy_matrix(m);
  for (size_t y = 0; y < m->rows; ++y)
    copy->data[y]->data[y] = 1.0;
  return copy;
}
#+END_SRC

*** ~slice_column~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~
+ Output: a pointer to a copy of the given ~Matrix_double~ with column at ~x~ sliced

#+BEGIN_SRC c
Matrix_double *slice_column(Matrix_double *m, size_t x) {
  Matrix_double *sliced = copy_matrix(m);

  for (size_t row = 0; row < m->rows; row++) {
    Array_double *old_row = sliced->data[row];
    sliced->data[row] = slice_element(old_row, x);
    free_vector(old_row);
  }
  sliced->cols--;

  return sliced;
}
#+END_SRC

*** ~add_column~
+ Author: Elizabet Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ and a new vector representing the appended column ~x~
+ Output: a pointer to a copy of the given ~Matrix_double~ with a new column ~x~

#+BEGIN_SRC c
Matrix_double *add_column(Matrix_double *m, Array_double *v) {
  Matrix_double *pushed = copy_matrix(m);

  for (size_t row = 0; row < m->rows; row++) {
    Array_double *old_row = pushed->data[row];
    pushed->data[row] = add_element(old_row, v->data[row]);
    free_vector(old_row);
  }

  pushed->cols++;
  return pushed;
}
#+END_SRC

*** ~copy_matrix~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~
+ Output: a pointer to a copy of the given ~Matrix_double~

#+BEGIN_SRC c
Matrix_double *copy_matrix(Matrix_double *m) {
  Matrix_double *copy = InitMatrixWithSize(double, m->rows, m->cols, 0.0);
  for (size_t y = 0; y < copy->rows; y++) {
    free_vector(copy->data[y]);
    copy->data[y] = copy_vector(m->data[y]);
  }
  return copy;
}
#+END_SRC
  
*** ~free_matrix~
+ Author: Elizabeth Hunt
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~
+ Output: none.
+ Side Effects: frees memory reserved by a given ~Matrix_double~ and its member
  ~Array_double~ vectors describing its rows.

#+BEGIN_SRC c
void free_matrix(Matrix_double *m) {
  for (size_t y = 0; y < m->rows; ++y)
    free_vector(m->data[y]);
  free(m);
}
#+END_SRC

*** ~format_matrix_into~
+ Author: Elizabeth Hunt
+ Name: ~format_matrix_into~
+ Location: ~src/matrix.c~
+ Input: a pointer to a ~Matrix_double~ and a pointer to a c-string ~s~ to "print" the vector out
  into
+ Output: nothing.
+ Side effect: overwritten memory into ~s~

#+BEGIN_SRC c
void format_matrix_into(Matrix_double *m, char *s) {
  if (m->rows == 0)
    strcpy(s, "empty");

  for (size_t y = 0; y < m->rows; ++y) {
    char row_s[256];
    strcpy(row_s, "");

    format_vector_into(m->data[y], row_s);
    strcat(s, row_s);
  }
  strcat(s, "\n");
}
#+END_SRC
** Root Finding Methods
*** ~find_ivt_range~
+ Author: Elizabeth Hunt
+ Name: ~find_ivt_range~
+ Location: ~src/roots.c~
+ Input: a pointer to a oneary function taking a double and producing a double, the beginning point
  in $R$ to search for a range, a ~delta~ step that is taken, and a ~max_steps~ number of maximum
  iterations to perform.
+ Output: a pair of ~double~'s in an ~Array_double~ representing a closed closed interval ~[beginning, end]~

#+BEGIN_SRC c
// f is well defined at start_x + delta*n for all n on the integer range [0,
// max_iterations]
Array_double *find_ivt_range(double (*f)(double), double start_x, double delta,
                             size_t max_iterations) {
  double a = start_x;

  while (f(a) * f(a + delta) >= 0 && max_iterations > 0) {
    max_iterations--;
    a += delta;
  }

  double end = a + delta;
  double begin = a - delta;

  if (max_iterations == 0 && f(begin) * f(end) >= 0)
    return NULL;
  return InitArray(double, {begin, end});
}
#+END_SRC
*** ~bisect_find_root~
+ Author: Elizabeth Hunt
+ Name(s): ~bisect_find_root~
+ Input: a one-ary function taking a double and producing a double, a closed interval represented
  by ~a~ and ~b~: ~[a, b]~, a ~tolerance~ at which we return the estimated root once $b-a < \text{tolerance}$, and a
  ~max_iterations~ to break us out of a loop if we can never reach the ~tolerance~.
+ Output: a vector of size of 3, ~double~'s representing first the range ~[a,b]~ and then the midpoint,
  ~c~ of the range.
+ Description: recursively uses binary search to split the interval until we reach ~tolerance~. We
  also assume the function ~f~ is continuous on ~[a, b]~.

#+BEGIN_SRC c
// f is continuous on [a, b]
Array_double *bisect_find_root(double (*f)(double), double a, double b,
                               double tolerance, size_t max_iterations) {
  assert(a <= b);
  // guarantee there's a root somewhere between a and b by IVT
  assert(f(a) * f(b) < 0);

  double c = (1.0 / 2) * (a + b);
  if (b - a < tolerance || max_iterations == 0)
    return InitArray(double, {a, b, c});

  if (f(a) * f(c) < 0)
    return bisect_find_root(f, a, c, tolerance, max_iterations - 1);
  return bisect_find_root(f, c, b, tolerance, max_iterations - 1);
}
#+END_SRC
*** ~bisect_find_root_with_error_assumption~
+ Author: Elizabeth Hunt
+ Name: ~bisect_find_root_with_error_assumption~
+ Input: a one-ary function taking a double and producing a double, a closed interval represented
  by ~a~ and ~b~: ~[a, b]~, and a ~tolerance~ equivalent to the above definition in ~bisect_find_root~
+ Output: a ~double~ representing the estimated root
+ Description: using the bisection method we know that $e_k \le (\frac{1}{2})^k (b_0 - a_0)$. So we can
  calculate $k$ at the worst possible case (that the error is exactly the tolerance) to be
  $\frac{log(tolerance) - log(b_0 - a_0)}{log(\frac{1}{2})}$. We pass this value into the ~max_iterations~
  of ~bisect_find_root~ as above.
#+BEGIN_SRC c
double bisect_find_root_with_error_assumption(double (*f)(double), double a,
                                              double b, double tolerance) {
  assert(a <= b);

  uint64_t max_iterations =
      (uint64_t)ceil((log(tolerance) - log(b - a)) / log(1 / 2.0));

  Array_double *a_b_root = bisect_find_root(f, a, b, tolerance, max_iterations);
  double root = a_b_root->data[2];
  free_vector(a_b_root);

  return root;
}
#+END_SRC

*** ~fixed_point_iteration_method~
+ Author: Elizabeth Hunt
+ Name: ~fixed_point_iteration_method~
+ Location: ~src/roots.c~
+ Input: a pointer to a oneary function $f$ taking a double and producing a double of which we are
  trying to find a root, a guess $x_0$, and a function $g$ of the same signature of $f$ at which we
  "step" our guesses according to the fixed point iteration method: $x_k = g(x_{k-1})$. Additionally, a
  ~max_iterations~ representing the maximum number of "steps" to take before arriving at our
  approximation and a ~tolerance~ to return our root if it becomes within [0 - tolerance, 0 + tolerance].
+ Assumptions: $g(x)$ must be a function such that at the point $x^*$ (the found root) the derivative
  $|g'(x^*)| \lt 1$
+ Output: a double representing the found approximate root $\approx x^*$.

#+BEGIN_SRC c
double fixed_point_iteration_method(double (*f)(double), double (*g)(double),
                                    double x_0, double tolerance,
                                    size_t max_iterations) {
  if (max_iterations <= 0)
    return x_0;

  double root = g(x_0);
  if (tolerance >= fabs(f(root)))
    return root;

  return fixed_point_iteration_method(f, g, root, tolerance,
                                      max_iterations - 1);
}
#+END_SRC

*** ~fixed_point_newton_method~
+ Author: Elizabeth Hunt
+ Name: ~fixed_point_newton_method~
+ Location: ~src/roots.c~
+ Input: a pointer to a oneary function $f$ taking a double and producing a double of which we are
  trying to find a root and another pointer to a function fprime of the same signature, a guess $x_0$,
  and a ~max_iterations~ and ~tolerance~ as defined in the above method are required inputs.
+ Description: continually computes elements in the sequence $x_n = x_{n-1} - \frac{f(x_{n-1})}{f'p(x_{n-1})}$
+ Output: a double representing the found approximate root $\approx x^*$ recursively applied to the sequence
  given
#+BEGIN_SRC c
double fixed_point_newton_method(double (*f)(double), double (*fprime)(double),
                                 double x_0, double tolerance,
                                 size_t max_iterations) {
  if (max_iterations <= 0)
    return x_0;

  double root = x_0 - f(x_0) / fprime(x_0);
  if (tolerance >= fabs(f(root)))
    return root;

  return fixed_point_newton_method(f, fprime, root, tolerance,
                                   max_iterations - 1);
}
#+END_SRC

*** ~fixed_point_secant_method~
+ Author: Elizabeth Hunt
+ Name: ~fixed_point_secant_method~
+ Location: ~src/roots.c~
+ Input: a pointer to a oneary function $f$ taking a double and producing a double of which we are
  trying to find a root, a guess $x_0$ and $x_1$ in which a root lies between $[x_0, x_1]$; applying the
  sequence $x_n = x_{n-1} - f(x_{n-1}) \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}$.
  Additionally, a ~max_iterations~ and ~tolerance~ as defined in the above method are required
  inputs.
+ Output: a double representing the found approximate root $\approx x^*$ recursively applied to the sequence.
#+BEGIN_SRC c
double fixed_point_secant_method(double (*f)(double), double x_0, double x_1,
                                 double tolerance, size_t max_iterations) {
  if (max_iterations == 0)
    return x_1;

  double root = x_1 - f(x_1) * ((x_1 - x_0) / (f(x_1) - f(x_0)));

  if (tolerance >= fabs(f(root)))
    return root;

  return fixed_point_secant_method(f, x_1, root, tolerance, max_iterations - 1);
}
#+END_SRC
*** ~fixed_point_secant_bisection_method~
+ Author: Elizabeth Hunt
+ Name: ~fixed_point_secant_method~
+ Location: ~src/roots.c~
+ Input: a pointer to a oneary function $f$ taking a double and producing a double of which we are
  trying to find a root, a guess $x_0$, and a $x_1$ of which we define our first interval $[x_0, x_1]$.
  Then, we perform a single iteration of the ~fixed_point_secant_method~ on this interval; if it
  produces a root outside, we refresh the interval and root respectively with the given
  ~bisect_find_root~ method. Additionally, a ~max_iterations~ and ~tolerance~ as defined in the above method are required
  inputs.
+ Output: a double representing the found approximate root $\approx x^*$ continually applied with the
  constraints defined.

#+BEGIN_SRC c
double fixed_point_secant_bisection_method(double (*f)(double), double x_0,
                                           double x_1, double tolerance,
                                           size_t max_iterations) {
  double begin = x_0;
  double end = x_1;
  double root = x_0;

  while (tolerance < fabs(f(root)) && max_iterations > 0) {
    max_iterations--;

    double secant_root = fixed_point_secant_method(f, begin, end, tolerance, 1);

    if (secant_root < begin || secant_root > end) {
      Array_double *range_root = bisect_find_root(f, begin, end, tolerance, 1);

      begin = range_root->data[0];
      end = range_root->data[1];
      root = range_root->data[2];

      free_vector(range_root);
      continue;
    }

    root = secant_root;

    if (f(root) * f(begin) < 0)
      end = secant_root; // the root exists in [begin, secant_root]
    else
      begin = secant_root;
  }

  return root;
}
#+END_SRC

** Linear Routines
*** ~least_squares_lin_reg~
+ Author: Elizabeth Hunt
+ Name: ~least_squares_lin_reg~
+ Location: ~src/lin.c~
+ Input: two pointers to ~Array_double~'s whose entries correspond two ordered
  pairs in R^2
+ Output: a linear model best representing the ordered pairs via least squares
  regression
  
#+BEGIN_SRC c
Line *least_squares_lin_reg(Array_double *x, Array_double *y) {
  assert(x->size == y->size);

  uint64_t n = x->size;
  double sum_x = sum_v(x);
  double sum_y = sum_v(y);
  double sum_xy = v_dot_v(x, y);
  double sum_xx = v_dot_v(x, x);
  double denom = ((n * sum_xx) - (sum_x * sum_x));

  Line *line = malloc(sizeof(Line));
  line->m = ((sum_xy * n) - (sum_x * sum_y)) / denom;
  line->a = ((sum_y * sum_xx) - (sum_x * sum_xy)) / denom;

  return line;
}
#+END_SRC

** Eigen-Adjacent
*** ~dominant_eigenvalue~
+ Author: Elizabeth Hunt
+ Name: ~dominant_eigenvalue~
+ Location: ~src/eigen.c~
+ Input: a pointer to an invertible matrix ~m~, an initial eigenvector guess ~v~ (that is non
  zero or orthogonal to an eigenvector with the dominant eigenvalue), a ~tolerance~ and
  ~max_iterations~ that act as stop conditions
+ Output: the dominant eigenvalue with the highest magnitude, approximated with the Power
  Iteration Method

#+BEGIN_SRC c
double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
                           size_t max_iterations) {
  assert(m->rows == m->cols);
  assert(m->rows == v->size);

  double error = tolerance;
  size_t iter = max_iterations;
  double lambda = 0.0;
  Array_double *eigenvector_1 = copy_vector(v);

  while (error >= tolerance && (--iter) > 0) {
    Array_double *eigenvector_2 = m_dot_v(m, eigenvector_1);
    Array_double *normalized_eigenvector_2 =
        scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2));
    free_vector(eigenvector_2);
    eigenvector_2 = normalized_eigenvector_2;

    Array_double *mx = m_dot_v(m, eigenvector_2);
    double new_lambda =
        v_dot_v(mx, eigenvector_2) / v_dot_v(eigenvector_2, eigenvector_2);

    error = fabs(new_lambda - lambda);
    lambda = new_lambda;
    free_vector(eigenvector_1);
    eigenvector_1 = eigenvector_2;
  }

  return lambda;
}
#+END_SRC
*** ~shift_inverse_power_eigenvalue~
+ Author: Elizabeth Hunt
+ Name: ~least_dominant_eigenvalue~
+ Location: ~src/eigen.c~
+ Input: a pointer to an invertible matrix ~m~, an initial eigenvector guess ~v~ (that is non
  zero or orthogonal to an eigenvector with the dominant eigenvalue), a ~shift~ to act as the
  shifted \delta, and ~tolerance~ and ~max_iterations~ that act as stop conditions.
+ Output: the eigenvalue closest to ~shift~ with the lowest magnitude closest to 0, approximated
  with the Inverse Power Iteration Method
#+BEGIN_SRC c
double shift_inverse_power_eigenvalue(Matrix_double *m, Array_double *v,
                                      double shift, double tolerance,
                                      size_t max_iterations) {
  assert(m->rows == m->cols);
  assert(m->rows == v->size);

  Matrix_double *m_c = copy_matrix(m);
  for (size_t y = 0; y < m_c->rows; ++y)
    m_c->data[y]->data[y] = m_c->data[y]->data[y] - shift;

  double error = tolerance;
  size_t iter = max_iterations;
  double lambda = shift;
  Array_double *eigenvector_1 = copy_vector(v);

  while (error >= tolerance && (--iter) > 0) {
    Array_double *eigenvector_2 = solve_matrix_lu_bsubst(m_c, eigenvector_1);
    Array_double *normalized_eigenvector_2 =
        scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2));
    free_vector(eigenvector_2);

    Array_double *mx = m_dot_v(m, normalized_eigenvector_2);
    double new_lambda =
        v_dot_v(mx, normalized_eigenvector_2) /
        v_dot_v(normalized_eigenvector_2, normalized_eigenvector_2);

    error = fabs(new_lambda - lambda);
    lambda = new_lambda;
    free_vector(eigenvector_1);
    eigenvector_1 = normalized_eigenvector_2;
  }

  return lambda;
}
#+END_SRC

*** ~least_dominant_eigenvalue~
+ Author: Elizabeth Hunt
+ Name: ~least_dominant_eigenvalue~
+ Location: ~src/eigen.c~
+ Input: a pointer to an invertible matrix ~m~, an initial eigenvector guess ~v~ (that is non
  zero or orthogonal to an eigenvector with the dominant eigenvalue), a ~tolerance~ and
  ~max_iterations~ that act as stop conditions.
+ Output: the least dominant eigenvalue with the lowest magnitude closest to 0, approximated
  with the Inverse Power Iteration Method.
#+BEGIN_SRC c
double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
                                 double tolerance, size_t max_iterations) {
  return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations);
}
#+END_SRC
*** ~partition_find_eigenvalues~
+ Author: Elizabeth Hunt
+ Name: ~partition_find_eigenvalues~
+ Location: ~src/eigen.c~
+ Input: a pointer to an invertible matrix ~m~, a matrix whose rows correspond to initial
  eigenvector guesses at each "partition" which is computed from a uniform distribution
  between the number of rows this "guess matrix" has and the distance between the least
  dominant eigenvalue and the most dominant. Additionally, a ~max_iterations~ and a ~tolerance~
  that act as stop conditions.
+ Output: a vector of ~doubles~ corresponding to the "nearest" eigenvalue at the midpoint of
  each partition, via the given guess of that partition.
#+BEGIN_SRC c
Array_double *partition_find_eigenvalues(Matrix_double *m,
                                         Matrix_double *guesses,
                                         double tolerance,
                                         size_t max_iterations) {
  assert(guesses->rows >=
         2); // we need at least, the most and least dominant eigenvalues

  double end = dominant_eigenvalue(m, guesses->data[guesses->rows - 1],
                                   tolerance, max_iterations);
  double begin =
      least_dominant_eigenvalue(m, guesses->data[0], tolerance, max_iterations);

  double delta = (end - begin) / guesses->rows;
  Array_double *eigenvalues = InitArrayWithSize(double, guesses->rows, 0.0);
  for (size_t i = 0; i < guesses->rows; i++) {
    double box_midpoint = ((delta * i) + (delta * (i + 1))) / 2;

    double nearest_eigenvalue = shift_inverse_power_eigenvalue(
        m, guesses->data[i], box_midpoint, tolerance, max_iterations);

    eigenvalues->data[i] = nearest_eigenvalue;
  }

  return eigenvalues;
}
#+END_SRC
*** ~leslie_matrix~
+ Author: Elizabeth Hunt
+ Name: ~leslie_matrix~
+ Location: ~src/eigen.c~
+ Input: two pointers to ~Array_double~'s representing the ratio of individuals in an age class
  $x$ getting to the next age class $x+1$ and the number of offspring that individuals in an age
  class create in age class 0.
+ Output: the leslie matrix generated from the input vectors.

#+BEGIN_SRC c
Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
                             Array_double *age_class_offspring) {
  assert(age_class_surivor_ratio->size + 1 == age_class_offspring->size);

  Matrix_double *leslie = InitMatrixWithSize(double, age_class_offspring->size,
                                             age_class_offspring->size, 0.0);

  free_vector(leslie->data[0]);
  leslie->data[0] = age_class_offspring;

  for (size_t i = 0; i < age_class_surivor_ratio->size; i++)
    leslie->data[i + 1]->data[i] = age_class_surivor_ratio->data[i];
  return leslie;
}
#+END_SRC

** Appendix / Miscellaneous
*** Data Types
**** ~Line~
+ Author: Elizabeth Hunt
+ Location: ~inc/types.h~

#+BEGIN_SRC c
typedef struct Line {
  double m;
  double a;
} Line;
#+END_SRC
**** The ~Array_<type>~ and ~Matrix_<type>~
+ Author: Elizabeth Hunt
+ Location: ~inc/types.h~

We define two Pre processor Macros ~DEFINE_ARRAY~ and ~DEFINE_MATRIX~ that take
as input a type, and construct a struct definition for the given type for
convenient access to the vector or matrices dimensions.

Such that ~DEFINE_ARRAY(int)~ would expand to:

#+BEGIN_SRC c
  typedef struct {
    int* data;
    size_t size;
  } Array_int
#+END_SRC

And ~DEFINE_MATRIX(int)~ would expand a to ~Matrix_int~; containing a pointer to
a collection of pointers of ~Array_int~'s and its dimensions.

#+BEGIN_SRC c
  typedef struct {
    Array_int **data;
    size_t cols;
    size_t rows;
  } Matrix_int
#+END_SRC

*** Macros
**** ~c_max~ and ~c_min~
+ Author: Elizabeth Hunt
+ Location: ~inc/macros.h~
+ Input: two structures that define an order measure
+ Output: either the larger or smaller of the two depending on the measure

#+BEGIN_SRC c
#define c_max(x, y) (((x) >= (y)) ? (x) : (y))
#define c_min(x, y) (((x) <= (y)) ? (x) : (y))
#+END_SRC

**** ~InitArray~
+ Author: Elizabeth Hunt
+ Location: ~inc/macros.h~
+ Input: a type and array of values to initialze an array with such type
+ Output: a new ~Array_type~ with the size of the given array and its data

#+BEGIN_SRC c
#define InitArray(TYPE, ...)                            \
  ({                                                    \
    TYPE temp[] = __VA_ARGS__;                          \
    Array_##TYPE *arr = malloc(sizeof(Array_##TYPE));   \
    arr->size = sizeof(temp) / sizeof(temp[0]);         \
    arr->data = malloc(arr->size * sizeof(TYPE));       \
    memcpy(arr->data, temp, arr->size * sizeof(TYPE));  \
    arr;                                                \
  })
#+END_SRC

**** ~InitArrayWithSize~
+ Author: Elizabeth Hunt
+ Location: ~inc/macros.h~
+ Input: a type, a size, and initial value
+ Output: a new ~Array_type~ with the given size filled with the initial value

#+BEGIN_SRC c
#define InitArrayWithSize(TYPE, SIZE, INIT_VALUE)      \
  ({                                                   \
    Array_##TYPE *arr = malloc(sizeof(Array_##TYPE));  \
    arr->size = SIZE;                                  \
    arr->data = malloc(arr->size * sizeof(TYPE));      \
    for (size_t i = 0; i < arr->size; i++)             \
      arr->data[i] = INIT_VALUE;                       \
    arr;                                               \
  })
#+END_SRC

**** ~InitMatrixWithSize~
+ Author: Elizabeth Hunt
+ Location: ~inc/macros.h~
+ Input: a type, number of rows, columns, and initial value
+ Output: a new ~Matrix_type~ of size ~rows x columns~ filled with the initial
  value

#+BEGIN_SRC c
#define InitMatrixWithSize(TYPE, ROWS, COLS, INIT_VALUE)                       \
  ({                                                                           \
    Matrix_##TYPE *matrix = malloc(sizeof(Matrix_##TYPE));                     \
    matrix->rows = ROWS;                                                       \
    matrix->cols = COLS;                                                       \
    matrix->data = malloc(matrix->rows * sizeof(Array_##TYPE *));              \
    for (size_t y = 0; y < matrix->rows; y++)                                  \
      matrix->data[y] = InitArrayWithSize(TYPE, COLS, INIT_VALUE);             \
    matrix;                                                                    \
  })
#+END_SRC

