The classical Gauss-Jordan method for matrix inversion involves augmenting the matrix with a unit matrix and requires a workspace twice as large as the original matrix as well as computational operations to be perform...The classical Gauss-Jordan method for matrix inversion involves augmenting the matrix with a unit matrix and requires a workspace twice as large as the original matrix as well as computational operations to be performed on both the original and the unit matrix. A modified version of the method for performing the inversion without explicitly generating the unit matrix by replicating its functionality within the original matrix space for more efficient utilization of computational resources is presented in this article. Although the algorithm described here picks the pivots solely from the diagonal which, therefore, may not contain a zero, it did not pose any problem for the author because he used it to invert structural stiffness matrices which met this requirement. Techniques such as row/column swapping to handle off-diagonal pivots are also applicable to this method but are beyond the scope of this article.展开更多
A modified version of the Gauss-Jordan algorithm for performing In-Place matrix inversion without using an augmenting unit matrix was described in a previous article by the author. He had also developed several Struct...A modified version of the Gauss-Jordan algorithm for performing In-Place matrix inversion without using an augmenting unit matrix was described in a previous article by the author. He had also developed several Structural Engineering softwares during his career using that method as their analysis engine. He chose matrix inversion because it was suitable for in-core solution of large numbers of vectors for the same set of equations as encountered in structural analysis of moving, dynamic and seismic loadings. The purpose of this article is to provide its readers with its theoretical background and detailed computations of an In-Place matrix inversion task as well as a Visual Basic routine of the algorithm for direct incorporation into Visual Basic 6TM softwares and Visual Basic for ApplicationsTM macros in MS-ExcelTM spreadsheets to save them time and effort of software development.展开更多
文摘The classical Gauss-Jordan method for matrix inversion involves augmenting the matrix with a unit matrix and requires a workspace twice as large as the original matrix as well as computational operations to be performed on both the original and the unit matrix. A modified version of the method for performing the inversion without explicitly generating the unit matrix by replicating its functionality within the original matrix space for more efficient utilization of computational resources is presented in this article. Although the algorithm described here picks the pivots solely from the diagonal which, therefore, may not contain a zero, it did not pose any problem for the author because he used it to invert structural stiffness matrices which met this requirement. Techniques such as row/column swapping to handle off-diagonal pivots are also applicable to this method but are beyond the scope of this article.
文摘A modified version of the Gauss-Jordan algorithm for performing In-Place matrix inversion without using an augmenting unit matrix was described in a previous article by the author. He had also developed several Structural Engineering softwares during his career using that method as their analysis engine. He chose matrix inversion because it was suitable for in-core solution of large numbers of vectors for the same set of equations as encountered in structural analysis of moving, dynamic and seismic loadings. The purpose of this article is to provide its readers with its theoretical background and detailed computations of an In-Place matrix inversion task as well as a Visual Basic routine of the algorithm for direct incorporation into Visual Basic 6TM softwares and Visual Basic for ApplicationsTM macros in MS-ExcelTM spreadsheets to save them time and effort of software development.
