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| − | '''Strassen algorithm for a symmetric matrix:'''
| + | === Theorem for the Strassen algorithm === |
| | + | Computing the [[w:Matrix multiplication|<span class="wikipedia">matrix product</span>]] <math>AA^T</math> shortens the original [[w:Execution_(computing)#Runtime|<span class="wikipedia">runtime</span>]] <math>T(n) = \mathcal{O}(n^{(_2 7)})</math> of the [[w:Strassen algorithm|<span class="wikipedia">Strassen algorithm</span>]] roughly by <math>\tilde{3}</math> for sufficiently big <math>n := 2^k, k \in \mathbb{N}^*</math> and the [[w:Matrix (mathematics)|<span class="wikipedia">matrix</span>]] <math>A \in \mathbb{C}^{n \times n}</math> due to the [[w:geometric series|<span class="wikipedia">geometric series</span>]] and |
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| − | For a [[w:Symmetric matrix|<span class="wikipedia">symmetric matrix</span>]] <math>A \in \mathbb{C}^{n \times n}</math> where <math>n \in \mathbb{N}^*</math>, the [[w:Runtime (program lifecycle phase)|<span class="wikipedia">runtime</span>]] <math>T_s(n)</math> of the [[w:Strassen algorithm|<span class="wikipedia">Strassen algorithm</span>]] for the [[w:Matrix multiplication|<span class="wikipedia">matrix product</span>]] <math>A^2</math> is about half that of the original algorithm in <math>\mathcal{O}(n^{(_2 7)})</math>.
| + | <div style="text-align:center;"><math>A := |
| − | | |
| − | '''Proof:''' For <math>A :=
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| − | \begin{pmatrix}
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| − | A_{11} & A_{12} \\
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| − | A_{12} & A_{22}
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| − | \end{pmatrix}</math>, it holds that <math>A^TA =
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| − | \begin{pmatrix}
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| − | A_{11}^TA_{11}+A_{12}^TA_{12} & A_{11}^TA_{12}+A_{12}^TA_{22} \\
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| − | A_{12}^TA_{11}+A_{22}^TA_{12} & A_{12}^TA_{12}+A_{22}^TA_{22}
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| − | \end{pmatrix}</math> and <math>T_s(2n) = 3T_s(n) + 2n^{(_2 7)}</math>. Thus <math>T_s(n) = 3T_s(n/2) + 2(n/2)^{(_2 7)}</math> and <math>T_s(n/2) = 3T_s(n/4) + 2(n/4)^{(_2 7)}</math>.
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| − | The [[w:Geometric series|<span class="wikipedia">geometric series</span>]] yields because of <math>T_s(1) = 1</math>: <math>T_s(n) = 27T_s(n/8) + 2/7n^{(_2 7)}(1+3/7 + (3/7)^2 + ...) = 3^{(_2n)} + 2/7n^{(_2 7)} (1-(3/7)^{(_2n)})/(1-3/7)</math> <math>= n^{(_2 3)} + \hat{2}(n^{(_2 7)}-n^{(_2 3)}) = \hat{2} (n^{(_2 3)} + n^{(_2 7)})</math>.<math>\square</math>
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| − | '''Strassen algorithm for a square matrix:'''
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| − | For a square [[w:Matrix (mathematics)|<span class="wikipedia">matrix</span>]] <math>A \in \mathbb{C}^{n \times n}</math> where <math>n \in \mathbb{N}^*</math>, the runtime <math>T_q(n)</math> of the Strassen algorithm is for the matrix product <math>A^TA</math> about <math>4/7</math> that of the original algorithm in <math>\mathcal{O}(n^{(_2 7)})</math>.
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| − | '''Proof:''' For <math>A :=
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| | \begin{pmatrix} | | \begin{pmatrix} |
| | A_{11} & A_{12} \\ | | A_{11} & A_{12} \\ |
| | A_{21} & A_{22} | | A_{21} & A_{22} |
| − | \end{pmatrix}</math>, it holds that <math>A^TA = | + | \end{pmatrix}</math> as well as <math>AA^T = |
| − | \begin{pmatrix}
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| − | A_{11}^TA_{11}+A_{21}^TA_{21} & A_{11}^TA_{12}+A_{21}^TA_{22} \\
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| − | A_{12}^TA_{11}+A_{22}^TA_{21} & A_{12}^TA_{12}+A_{22}^TA_{22}
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| − | \end{pmatrix}</math> such that <math>T_q(2n) = 4T_s(n) + 2n^{(_2 7)}</math> and <math>T_q(n) = 4T_s(n/2) + 2/7n^{(_2 7)} = 2/3n^{(_2 3)} + 4/7n^{(_2 7)}</math>.<math>\square</math>
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| − | '''New algorithm for two symmetric matrices:'''
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| − | For two symmetric matrices <math>A, B \in \mathbb{C}^{n \times n}</math> where <math>n \in \mathbb{N}^*</math> the runtime is for the matrix product <math>AB</math> about <math>6/7</math> that of the original algorithm in <math>\mathcal{O}(n^{(_2 7)})</math>.
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| − | '''Proof:''' For <math>B :=
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| | \begin{pmatrix} | | \begin{pmatrix} |
| − | B_{11} & B_{12} \\ | + | A_{11}A_{11}^T+A_{12}A_{12}^T & A_{11}A_{21}^T+A_{12}A_{22}^T \\ |
| − | B_{12} & B_{22}
| + | A_{21}A_{11}^T+A_{22}A_{12}^T & A_{21}A_{21}^T+A_{22}A_{22}^T |
| − | \end{pmatrix}</math>, it holds that <math>A^TB =
| + | \end{pmatrix}.\square</math></div> |
| − | \begin{pmatrix}
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| − | A_{11}^TB_{11}+A_{12}^TB_{12} & A_{11}^TB_{12}+A_{12}^TB_{22} \\
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| − | A_{12}^TB_{11}+A_{22}^TB_{12} & A_{12}^TB_{12}+A_{22}^TB_{22} | |
| − | \end{pmatrix} =: C</math>. Putting
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| − | : <math>M_{1} := A_{11} \cdot (B_{11} + B_{12})</math>
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| − | : <math>M_{2} := A_{22} \cdot (B_{12} + B_{22})</math>
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| − | : <math>M_{3} := A_{12} \cdot (B_{22} - B_{11})</math>
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| − | : <math>M_{4} := (A_{12} - A_{11})\cdot B_{12}</math>
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| − | : <math>M_{5} := (A_{12} - A_{22})\cdot B_{22}</math>
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| − | : <math>M_{6} := (A_{12} - A_{11}) \cdot B_{11}</math>
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| − | implies
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| − | : <math>C_{11} = M_{1} + M_{4}</math>
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| − | : <math>C_{12} = M_{1} + M_{3} + M_{6}</math>
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| − | : <math>C_{21} = M_{2} - M_{3} + M_{5}</math>
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| − | : <math>C_{22} = C_{12} + M_{4} - M_{5}.\square</math>
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| − | '''New algorithm for a symmetric matrix:'''
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| − | For a symmetric matrix <math>A \in \mathbb{C}^{n \times n}</math> where <math>n \in \mathbb{N}^*</math> the runtime <math>T_p(n)</math> of the new algorithm is for the matrix product <math>A^2</math> by computing analogously as above about <math>3/7</math> that of the original algorithm in <math>\mathcal{O}(n^{(_2 7)}).\square</math>
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| − | '''New algorithm for a square matrix:'''
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| − | For a square matrix <math>A \in \mathbb{C}^{n \times n}</math> where <math>n \in \mathbb{N}^*</math> the runtime <math>T_r(n)</math> of the new algorithm is for the matrix product <math>A^TA</math> about <math>24/49</math> that of the original algorithm (see above) in <math>\mathcal{O}(n^{(_2 7)}).\square</math>
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| | == See also == | | == See also == |