diff --git a/Diagram_overleaf/Errors_vs_l.png b/Diagram_overleaf/Errors_vs_l.png
new file mode 100644
index 0000000..439668d
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diff --git a/Diagram_overleaf/Numerical Stability.png b/Diagram_overleaf/Numerical Stability.png
new file mode 100644
index 0000000..96aef12
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diff --git a/Diagram_overleaf/convergence.png b/Diagram_overleaf/convergence.png
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diff --git a/README.md b/README.md
index 467d5f3..508b3bd 100644
--- a/README.md
+++ b/README.md
@@ -46,7 +46,7 @@ The complete algorithm involves the following steps:
    - These values are used to determine the function $f(\tau, B)$, which is central to the fixed-point iteration scheme.
 
 7. **Compute the Derivative of $f(\tau_i, B)$**:
-   - We approximate the derivative $f'(\tau, B)$ using finite differences.
+   - We use the closed form solution for Derivative of f 
    - This derivative is critical for the Jacobi-Newton update step.
 
 8. **Update Boundary Values Using Jacobi-Newton Scheme**:
diff --git a/Testing.ipynb b/Testing.ipynb
index af37eb4..a7d3d58 100644
--- a/Testing.ipynb
+++ b/Testing.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -12,30 +12,32 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [],
    "source": [
-    "test1 = AmericanOptionPricing(100, 0.05, 0.05, 0.25, 1, 25, 4, 9, 51, option_type=OptionType.Put,eta=0.8)"
+    "test1 = AmericanOptionPricing(100, 0.05, 0.05, 0.25, 1, 25, 5, 12, 61, option_type=OptionType.Put,eta=0.8)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Starting iteration 1/4\n",
-      "Iteration 1/4 completed.\n",
-      "Starting iteration 2/4\n",
-      "Iteration 2/4 completed.\n",
-      "Starting iteration 3/4\n",
-      "Iteration 3/4 completed.\n",
-      "Starting iteration 4/4\n",
-      "Iteration 4/4 completed.\n",
+      "Starting iteration 1/5\n",
+      "Iteration 1/5 completed.\n",
+      "Starting iteration 2/5\n",
+      "Iteration 2/5 completed.\n",
+      "Starting iteration 3/5\n",
+      "Iteration 3/5 completed.\n",
+      "Starting iteration 4/5\n",
+      "Iteration 4/5 completed.\n",
+      "Starting iteration 5/5\n",
+      "Iteration 5/5 completed.\n",
       "Jacobi-Newton iterations completed.\n"
      ]
     }
@@ -47,35 +49,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "array([ 61.24225892,  61.27712165,  61.38204477,  61.55803   ,\n",
-       "        61.80673573,  62.13045962,  62.53211128,  63.01517446,\n",
-       "        63.58365337,  64.24200295,  64.995034  ,  65.84779433,\n",
-       "        66.80541202,  67.87290423,  69.05493176,  70.35550743,\n",
-       "        71.77763053,  73.32286494,  74.99082224,  76.77858653,\n",
-       "        78.68002539,  80.68506055,  82.77881187,  84.940758  ,\n",
-       "        87.14375634,  89.35320157,  91.52597224,  93.60973108,\n",
-       "        95.54162409,  97.24761131,  98.63915229,  99.60889207,\n",
-       "       100.        ])"
-      ]
-     },
-     "execution_count": 15,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "test1.updateded_boundary"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -84,16 +58,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "(9.462492596167081, 0.10696407630320409)"
+       "(9.462492596167081, 0.10695523378736016)"
       ]
      },
-     "execution_count": 37,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }