QR_Inverse

Overview

QR_Inverse, matrix inversion with the usage of QR decomposition. As matrix \(A\) could be decomposed into a product of an orthogonal matrix \(Q\) and an upper triangular matrix \(R\) in the form of \(A = QR\), the matrix inversion should be \(A^{-1} = R^{-1}Q^{-1} = R^{-1}Q^T\).

\[ \begin{align}\begin{aligned}AA^{-1} = I\\A = QR\\A^{-1} = R^{-1}Q^T\end{aligned}\end{align} \]

As matrix \(R\) is an upper triangular matrix, \(R^{-1}\) is easy to compute. In this design, \(R^{-1}\) is computed via Backwark Subsititution.

Implementation

DataType Supported

• float
• x_complex<float>
• std::complex<float>

Note

Subnormall values are not supported. If used, the synthesized hardware will flush these to zero, and the behavior will differ versus software simulation.

Interfaces

• Template parameters:
  • RowsColsA: Defines the matrix dimensions
  • InputType: Input data type
  • OutputType: Output data type
  • QRInverseTraits: QRInverse Traits class
• Arguments:
  • matrixAStrm: Stream of Input matrix A
  • matrixInverseAStrm: Stream of Inverse of input matrix
  • A_singular: 1 = Failure, matrix A is singular; 0 = success

Implementation Controls

Specifications

The DATAFLOW directive is applied to the top function. User could specify the individual sub-function implementiontations using a configuration class derived from the following basic class by redefining the appropriate class member:

template <int RowsColsA, typename InputType, typename OutputType>
struct qrInverseTraits {
    typedef float InternalType;
    typedef qrfTraits QRF_CONFIG;
    typedef backSubstituteTraits<RowsColsA, InternalType, InternalType> BACK_SUB_CONFIG;
    typedef matrixMultiplyTraits<NoTranspose,
                                 NoTranspose,
                                 RowsColsA,
                                 RowsColsA,
                                 RowsColsA,
                                 RowsColsA,
                                 InternalType,
                                 OutputType>
        MULTIPLIER_CONFIG;
};

The configuration class is supplied to the xf::solver::qrInverse function as a template paramter as follows. The sub-functions are executed sequentially: QRF, back substitution, and matrix multiply. The implementation selected for these sub-functions determines the resource utilization and function throughput/latency of the Inverse function.

template <int RowsColsA,
          typename InputType,
          typename OutputType,
          typename QRInverseTraits = qrInverseTraits<RowsColsA, InputType, OutputType> >
void qrInverse(hls::stream<InputType>& matrixAStrm, hls::stream<OutputType>& matrixInverseAStrm, int& A_singular) {
#pragma HLS DATAFLOW
    // Define intermediate buffers
    hls::stream<typename QRInverseTraits::InternalType> matrixQStrm;
#pragma HLS STREAM variable = matrixQStrm depth = 16
    hls::stream<typename QRInverseTraits::InternalType> matrixRStrm;
#pragma HLS STREAM variable = matrixRStrm depth = 16
    hls::stream<typename QRInverseTraits::InternalType> matrixInverseRStrm;
#pragma HLS STREAM variable = matrixInverseRStrm depth = 16

    // Run QR factorization, get upper-triangular result in R, orthogonal/unitary matrix Q
    const bool TRANSPOSED_Q = true; // Q is produced in transpose form such that Q*A = R
    qrf<TRANSPOSED_Q, RowsColsA, RowsColsA, InputType, typename QRInverseTraits::InternalType, typename QRInverseTraits::QRF_CONFIG>(matrixAStrm, matrixQStrm, matrixRStrm);

     // Run back-substitution to compute R^-1
     backSubstitute<RowsColsA, typename QRInverseTraits::InternalType, typename QRInverseTraits::InternalType, typename QRInverseTraits::BACK_SUB_CONFIG>(matrixRStrm, matrixInverseRStrm, A_singular);

     // A^-1 = R^-1*Qt
     matrixMultiply<NoTranspose, NoTranspose, RowsColsA, RowsColsA, RowsColsA, RowsColsA, RowsColsA, RowsColsA, typename QRInverseTraits::InternalType, OutputType, typename QRInverseTraits::MULTIPLIER_CONFIG>(matrixInverseRStrm, matrixQStrm, matrixInverseAStrm);
 }