diff --git a/inc/lizfcm.h b/inc/lizfcm.h
index 99acbf2..40046d2 100644
--- a/inc/lizfcm.h
+++ b/inc/lizfcm.h
@@ -30,7 +30,9 @@ extern Line *least_squares_lin_reg(Array_double *x, Array_double *y);
 extern void put_identity_diagonal(Matrix_double *m);
 extern Matrix_double *copy_matrix(Matrix_double *m);
 extern void free_matrix(Matrix_double *m);
-extern Matrix_double **put_lu_decomp(Matrix_double *m);
+extern Matrix_double **lu_decomp(Matrix_double *m);
+extern Array_double *bsubst(Matrix_double *u, Array_double *b);
+extern Array_double *fsubst(Matrix_double *l, Array_double *b);
 
 extern void format_matrix_into(Matrix_double *m, char *s);
 
diff --git a/lib/lizfcm.a b/lib/lizfcm.a
index 78ca79e..7ea59ed 100644
Binary files a/lib/lizfcm.a and b/lib/lizfcm.a differ
diff --git a/notes/Oct-13.org b/notes/Oct-13.org
new file mode 100644
index 0000000..853a6d6
--- /dev/null
+++ b/notes/Oct-13.org
@@ -0,0 +1,8 @@
+* Root Finding
+Finx x \in R such that f(x) = 0
+
+If g(x) is a function and we want to find x such that g(x) is
+extremal then we find x \in R such that g'(x) = 0
+
+
+Hanging Cable Problem y(x) = c_1 cosh(\frac{x- c_2}{c_2})
diff --git a/src/matrix.c b/src/matrix.c
index 953d50f..438a468 100644
--- a/src/matrix.c
+++ b/src/matrix.c
@@ -1,7 +1,7 @@
 #include "lizfcm.h"
 #include <assert.h>
-#include <string.h>
 #include <stdio.h>
+#include <string.h>
 
 void put_identity_diagonal(Matrix_double *m) {
   assert(m->rows == m->cols);
@@ -19,7 +19,7 @@ Matrix_double *copy_matrix(Matrix_double *m) {
   return copy;
 }
 
-Matrix_double **put_lu_decomp(Matrix_double *m) {
+Matrix_double **lu_decomp(Matrix_double *m) {
   assert(m->cols == m->rows);
 
   Matrix_double *u = copy_matrix(m);
@@ -63,6 +63,32 @@ Matrix_double **put_lu_decomp(Matrix_double *m) {
   return u_l;
 }
 
+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;
+}
+
+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;
+}
+
 void free_matrix(Matrix_double *m) {
   for (size_t y = 0; y < m->rows; ++y)
     free_vector(m->data[y]);
diff --git a/test/main.c b/test/main.c
index 75e60ee..2364b3d 100644
--- a/test/main.c
+++ b/test/main.c
@@ -10,19 +10,18 @@ int main() {
   printf("dmaceps(): %.10e\n", dmaceps());
   printf("========\n");
 
+  printf("Norm, Distance\n");
   Array_double *v = InitArray(double, {3, 1, -4, 1, 5, -9, 3});
-  char v_s[256];
-  strcpy(v_s, "");
-  format_vector_into(v, v_s);
+  char s[2048];
+  strcpy(s, "");
+  format_vector_into(v, s);
+  printf("v: %s", s);
 
   Array_double *w = InitArray(double, {-2, 7, 1, -8, -2, 8, 5});
-  char w_s[256];
-  strcpy(w_s, "");
-  format_vector_into(w, w_s);
+  strcpy(s, "");
+  format_vector_into(w, s);
+  printf("w: %s", s);
 
-  printf("Norm, Distance\n");
-  printf("v: %s", v_s);
-  printf("w: %s", w_s);
   printf("l1_norm(v): %f\n", l1_norm(v));
   printf("l2_norm(v): %f\n", l2_norm(v));
   printf("linf_norm(v): %f\n", linf_norm(v));
@@ -38,52 +37,79 @@ int main() {
   printf("approx f'(1) w/ fw.d.: %f\n", forward_derivative_at(&f, 1, h));
   printf("approx f'(1) w/ bw.d.: %f\n", backward_derivative_at(&f, 1, h));
   printf("========\n");
+  printf("Least Squares\n");
 
   v = InitArray(double, {1, 2, 3, 4, 5});
-  strcpy(v_s, "");
-  format_vector_into(v, v_s);
+  strcpy(s, "");
+  format_vector_into(v, s);
+  printf("v: %s", s);
   w = InitArray(double, {2, 3, 4, 5, 6});
-  strcpy(w_s, "");
-  format_vector_into(w, w_s);
+  strcpy(s, "");
+  format_vector_into(w, s);
+  printf("w: %s", s);
+
   Line *line = least_squares_lin_reg(v, w);
-  printf("Least Squares\n");
-  printf("v: %s", v_s);
-  printf("w: %s", w_s);
+
   printf("least_squares_lin_reg(v, w): (%f)x + %f\n", line->m, line->a);
   v = InitArray(double, {1, 2, 3, 4, 5, 6, 7});
-  strcpy(v_s, "");
-  format_vector_into(v, v_s);
+  strcpy(s, "");
+  format_vector_into(v, s);
+  printf("v: %s", s);
   w = InitArray(double, {0.5, 3, 2, 3.5, 5, 6, 7.5});
-  strcpy(w_s, "");
-  format_vector_into(w, w_s);
-  printf("v: %s", v_s);
-  printf("w: %s", w_s);
+  strcpy(s, "");
+  format_vector_into(w, s);
+  printf("w: %s", s);
+
   line = least_squares_lin_reg(v, w);
   printf("least_squares_lin_reg(v, w): (%f)x + %f\n", line->m, line->a);
   printf("========\n");
 
   printf("LU Decomp\n");
-  char m_s[2048];
-  Matrix_double *m = InitMatrixWithSize(double, 8, 8, 0.0);
-  for (int i = 0; i < 8; i++) {
-    for (int j = 0; j < 8; j++)
-      m->data[i]->data[j] = (i + 1.0) + j * 3 + (rand() % 12);
+  Matrix_double *m = InitMatrixWithSize(double, 10, 10, 0.0);
+  for (int i = 0; i < 10; i++) {
+    for (int j = 0; j < 10; j++)
+      m->data[i]->data[j] = (100 - rand() % 200);
   }
-  format_matrix_into(m, m_s);
-  printf("m = %s", m_s);
+  strcpy(s, "");
+  format_matrix_into(m, s);
+  printf("m = %s", s);
 
-  Matrix_double **u_l = put_lu_decomp(m);
+  Array_double *b = InitArrayWithSize(double, 10, 100.0);
+  Matrix_double **u_l = lu_decomp(m);
   Matrix_double *u = u_l[0];
   Matrix_double *l = u_l[1];
 
-  strcpy(m_s, "");
-  format_matrix_into(u, m_s);
-  printf("u = %s", m_s);
-  strcpy(m_s, "");
-  format_matrix_into(l, m_s);
-  printf("l = %s", m_s);
+  strcpy(s, "");
+  format_matrix_into(u, s);
+  printf("u = %s", s);
+  strcpy(s, "");
+  format_matrix_into(l, s);
+  printf("l = %s", s);
+  strcpy(s, "");
+  format_vector_into(b, s);
+  printf("b = %s", s);
   printf("========\n");
-  printf("Back Substitution\n");
+  printf("Backward -> Forward Substitution\n");
+
+  Array_double *b_fsub = fsubst(l, b);
+  strcpy(s, "");
+  format_vector_into(b_fsub, s);
+  printf("x: %s\n", s);
+
+  Array_double *x_bsub = bsubst(u, b_fsub);
+  strcpy(s, "");
+  format_vector_into(x_bsub, s);
+  printf("x: %s", s);
+
+  free_vector(b_fsub);
+
+  printf("Verifications (each should be approximately 100)\n");
+  for (size_t row = 0; row < m->rows; row++) {
+    double curr = 0;
+    for (size_t col = 0; col < m->cols; col++)
+      curr += m->data[row]->data[col] * x_bsub->data[col];
+    printf("Substitution for row %zu = %f\n", row, curr);
+  }
 
   return 0;
 }