# Modified givens rotations for inverse updating in qr decomposition

### Modified givens rotations for inverse updating in qr decomposition - marques houston dating 2016

An embodiment relates generally to a method for nulling a cell of a complex matrix.The complex matrix and a modified Givens rotation matrix are obtained by a processing unit, and the processing unit multiplies the complex matrix with the modified Givens rotation matrix for the nulling of the cell of the complex matrix to provide a modified form thereof.

Accordingly, it would be desirable and useful to be able to perform a Givens rotation on a complex matrix directly, namely without having to perform a complex-to-real data reduction prior to performing a Givens rotation.

One or more aspects generally relate to a modified Givens rotation.

More particularly, one or more aspects relate to directly performing a Givens rotation on a complex matrix.

As is known, systolic arrays have been used for VLSI systems.

A systolic array is a matrix of individual signal processing cells, where overall operation of the systolic array depends upon functions of the individual signal processing cells and the interconnection scheme of such signal processing cells.

The modified Givens rotation matrix includes complex numbers c*, c, −s, and s*, wherein the complex number s* is the complex conjugate of the complex number s, and wherein the complex number c* is the complex conjugate of the complex number c.

(As used herein, “outputting” or “output” can mean, without limitation, writing to a memory or a file, passing to another cell or block, writing to a user display or other output device, playing audible notifications or providing other physical feedback to a user, sending or transmitting to another system, exporting, or the like.) In one or more other embodiments of a method for nulling a cell of a complex matrix, one or more of the following aspects may be included.

A boundary cell and an internal cell of a systolic array for processing real numbers are described in detail in U. Likewise, internal cell 710, in contrast to the internal cell described in Mc Whirter, has a φ input and a 0 output each of which is associated with the phase of a complex number. Likewise, lists equations 711 for implementing internal cell 710.

Another paper that disclosed factoring out the phase and applying it at a later stage is “Efficient Implementation of Rotation Operations for High Performance QRD-RLS Filtering”, by B. In contrast to such equations in Mc Whirter, equations 701 and 711 include having to deal with phase φ.

The modified Givens rotation matrix may be implemented as a systolic array or otherwise used for processing complex numbers or matrices.multiplying the complex matrix with the modified Givens rotation matrix by the processing unit for the nulling of the cell of the complex matrix to provide a modified form thereof, the modified form comprising real numbers; andthe modified Givens rotation matrix including complex numbers c*, c, −s, and s*, wherein the complex number s* is the complex conjugate of the complex number s, and wherein the complex number c* is the complex conjugate of the complex number c; and5.

A non-transitory machine-readable medium having stored thereon information representing instructions that, when executed by processor means, causes the processor means to perform operations for nulling a cell of a complex matrix, the operations when executed comprising: the modified Givens rotation matrix including complex numbers c*, c, −s, and s*, wherein the complex number s* is the complex conjugate of the complex number s, and wherein the complex number c* is the complex conjugate of the complex number c;wherein the modified Givens rotation matrix includes complex numbers c*, c, −s, and s*, wherein the complex number s* is the complex conjugate of the complex number s, and wherein the complex number c* is the complex conjugate of the complex number c; and, wherein the processing unit is further configured for repeating the act of multiplying with variations of the modified Givens rotation matrix responsive to cells to be nulled for triangularization of the complex matrix.

A complex matrix and a modified Givens rotation matrix are obtained for multiplication by a processing unit, such as a systolic array or a CPU, for example, for the nulling of the cell to provide a modified form of the complex matrix. The modified Givens rotation matrix includes complex numbers c*, c, −s, and s*, wherein the complex number s* is the complex conjugate of the complex number s, and wherein the complex number c* is the complex conjugate of the complex number c.