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Update tutorial3_orthogonalization.ipynb

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Deniz
2025-09-17 11:20:18 +03:00
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tutorials/tutorial3_orthogonalization.ipynb
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@@ -104,6 +104,34 @@
"⚠️ CGS can lose orthogonality in finite precision arithmetic.\n" "⚠️ CGS can lose orthogonality in finite precision arithmetic.\n"
] ]
}, },
{
"cell_type": "code",
"execution_count": 1,
"id": "3fe97ce3",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Q (CGS): Matrix([[0.8164965809277261, -0.5520524474738834], [0.4082482904638631, 0.7590721152765896], [0.4082482904638631, 0.34503277967117707]])\n",
"R (CGS): Matrix([[2.449489742783178, 0.4082482904638631], [0.0, 2.41522945769824]])\n"
]
}
],
"source": [
"from numethods import Matrix, Vector\n",
"from numethods import QRGramSchmidt, QRModifiedGramSchmidt, QRHouseholder, LeastSquaresSolver\n",
"\n",
"# Example matrix\n",
"A = Matrix([[2, -1], [1, 2], [1, 1]])\n",
"\n",
"# Classical Gram-Schmidt\n",
"qrg = QRGramSchmidt(A)\n",
"print(\"Q (CGS):\", qrg.Q)\n",
"print(\"R (CGS):\", qrg.R)"
]
},
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "ba84b59a", "id": "ba84b59a",
@@ -126,6 +154,28 @@
"MGS is more stable than CGS.\n" "MGS is more stable than CGS.\n"
] ]
}, },
{
"cell_type": "code",
"execution_count": 2,
"id": "e01e25ff",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Q (MGS): Matrix([[0.8164965809277261, -0.5520524474738834], [0.4082482904638631, 0.7590721152765896], [0.4082482904638631, 0.34503277967117707]])\n",
"R (MGS): Matrix([[2.449489742783178, 0.4082482904638631], [0.0, 2.41522945769824]])\n"
]
}
],
"source": [
"# Modified Gram-Schmidt\n",
"qrm = QRModifiedGramSchmidt(A)\n",
"print(\"Q (MGS):\", qrm.Q)\n",
"print(\"R (MGS):\", qrm.R)\n"
]
},
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "d893d189", "id": "d893d189",
@@ -152,6 +202,28 @@
"$$" "$$"
] ]
}, },
{
"cell_type": "code",
"execution_count": 3,
"id": "15dfc35c",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Q (Householder): Matrix([[-0.8164965809277258, 0.552052447473883, -0.16903085094570333], [-0.40824829046386296, -0.7590721152765892, -0.5070925528371099], [-0.40824829046386296, -0.34503277967117707, 0.8451542547285166]])\n",
"R (Householder): Matrix([[-2.449489742783177, -0.408248290463863], [2.220446049250313e-16, -2.415229457698238], [2.220446049250313e-16, 2.220446049250313e-16]])\n"
]
}
],
"source": [
"# Householder QR\n",
"qrh = QRHouseholder(A)\n",
"print(\"Q (Householder):\", qrh.Q)\n",
"print(\"R (Householder):\", qrh.R)\n"
]
},
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "2b6c612f", "id": "2b6c612f",
@@ -177,62 +249,25 @@
"So we can solve using back-substitution.\n" "So we can solve using back-substitution.\n"
] ]
}, },
{
"cell_type": "markdown",
"id": "2740a134",
"metadata": {},
"source": [
"\n",
"## 6. Examples with `numethods`\n"
]
},
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 3, "execution_count": 4,
"id": "212e6f58", "id": "25b399b7",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"name": "stdout", "name": "stdout",
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"Q (CGS): Matrix([[0.8164965809277261, -0.5520524474738834], [0.4082482904638631, 0.7590721152765896], [0.4082482904638631, 0.34503277967117707]])\n",
"R (CGS): Matrix([[2.449489742783178, 0.4082482904638631], [0.0, 2.41522945769824]])\n",
"Q (MGS): Matrix([[0.8164965809277261, -0.5520524474738834], [0.4082482904638631, 0.7590721152765896], [0.4082482904638631, 0.34503277967117707]])\n",
"R (MGS): Matrix([[2.449489742783178, 0.4082482904638631], [0.0, 2.41522945769824]])\n",
"Q (Householder): Matrix([[-0.8164965809277258, 0.552052447473883, -0.16903085094570333], [-0.40824829046386296, -0.7590721152765892, -0.5070925528371099], [-0.40824829046386296, -0.34503277967117707, 0.8451542547285166]])\n",
"R (Householder): Matrix([[-2.449489742783177, -0.408248290463863], [2.220446049250313e-16, -2.415229457698238], [2.220446049250313e-16, 2.220446049250313e-16]])\n",
"Least squares solution: Vector([1.0285714285714287, 0.828571428571429])\n" "Least squares solution: Vector([1.0285714285714287, 0.828571428571429])\n"
] ]
} }
], ],
"source": [ "source": [
"\n",
"from numethods import Matrix, Vector\n",
"from numethods import QRGramSchmidt, QRModifiedGramSchmidt, QRHouseholder, LeastSquaresSolver\n",
"\n",
"# Example matrix\n",
"A = Matrix([[2, -1], [1, 2], [1, 1]])\n",
"\n",
"# Classical Gram-Schmidt\n",
"qrg = QRGramSchmidt(A)\n",
"print(\"Q (CGS):\", qrg.Q)\n",
"print(\"R (CGS):\", qrg.R)\n",
"\n",
"# Modified Gram-Schmidt\n",
"qrm = QRModifiedGramSchmidt(A)\n",
"print(\"Q (MGS):\", qrm.Q)\n",
"print(\"R (MGS):\", qrm.R)\n",
"\n",
"# Householder QR\n",
"qrh = QRHouseholder(A)\n",
"print(\"Q (Householder):\", qrh.Q)\n",
"print(\"R (Householder):\", qrh.R)\n",
"\n",
"# Least squares example\n", "# Least squares example\n",
"b = Vector([1, 2, 3])\n", "b = Vector([1, 2, 3])\n",
"x_ls = LeastSquaresSolver(A, b).solve()\n", "x_ls = LeastSquaresSolver(A, b).solve()\n",
"print(\"Least squares solution:\", x_ls)\n" "print(\"Least squares solution:\", x_ls)"
] ]
}, },
{ {
@@ -241,7 +276,7 @@
"metadata": {}, "metadata": {},
"source": [ "source": [
"\n", "\n",
"## 7. Key Takeaways\n", "## 6. Key Takeaways\n",
"\n", "\n",
"- CGS is simple but numerically unstable.\n", "- CGS is simple but numerically unstable.\n",
"- MGS is more stable and preferred if using Gram-Schmidt.\n", "- MGS is more stable and preferred if using Gram-Schmidt.\n",