凸优化与数值线性代数：算法复杂性和求解技巧
这篇博客讨论了凸优化问题在数值线性代数中的应用，包括矩阵运算的浮点数运算次数、求解已因式分解的线性方程组的效率、LU和Cholesky分解以及分块消元和Schur补的计算复杂性。文章深入探讨了不同算法的时间预测，并分析了矩阵乘法顺序对运算次数的影响。
矩阵结果与算法复杂性
Algorithm flop counts allow for very accurate and precise prediction of running time on a given computer.
About how long does a 1 Gflop computer take to solve a system of 100 linear equations with 100 variables?
About how long does a 1 Gflop computer take to solve 10 systems of of 100 linear equations with 100 variables, with the same coefficient matrix but 10 different right hand sides?
Since matrix multiplication is associative, the flop count for multiplying three or more matrices doesn't depend on the order in which you multiply them.
多个矩阵相乘，计算的顺序对浮点数运算次数有影响，比如
考虑计算ABC的两种顺序，
（1）先计算AB，再计算(AB)C，首先计算AB所需浮点计算次数为mp(2n-1)，再计算(AB)C所需浮点计算次数为mq(2p-1)，一共mp(2n-1)+mq(2p-1)=2mp(n+q)-m(p+q)
（2）先计算BC，再计算A(BC)，首先计算BC所需浮点计算次数为nq(2p-1)，再计算A(BC)所需浮点计算次数为mq(2n-1)，一共nq(2p-1)+mq(2n-1)=2qn(m+p)-q(m+n)
求解已经因式分解的矩阵的线性方程组
Suppose is lower triangular. The flop count for computing Ab is the same order as the flop count for computing
.
下三角矩阵的逆矩阵也是下三角矩阵，所以计算Ab和计算所需要
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