692 lines
22 KiB
C
692 lines
22 KiB
C
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/* ----------------------------------------------------------------------
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* Project: CMSIS DSP Library
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* Title: arm_mat_inverse_f32.c
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* Description: Floating-point matrix inverse
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*
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* $Date: 27. January 2017
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* $Revision: V.1.5.1
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*
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* Target Processor: Cortex-M cores
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* -------------------------------------------------------------------- */
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/*
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* Copyright (C) 2010-2017 ARM Limited or its affiliates. All rights reserved.
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*
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* SPDX-License-Identifier: Apache-2.0
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*
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* Licensed under the Apache License, Version 2.0 (the License); you may
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* not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an AS IS BASIS, WITHOUT
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* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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#include "arm_math.h"
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/**
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* @ingroup groupMatrix
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*/
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/**
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* @defgroup MatrixInv Matrix Inverse
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*
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* Computes the inverse of a matrix.
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*
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* The inverse is defined only if the input matrix is square and non-singular (the determinant
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* is non-zero). The function checks that the input and output matrices are square and of the
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* same size.
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*
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* Matrix inversion is numerically sensitive and the CMSIS DSP library only supports matrix
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* inversion of floating-point matrices.
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*
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* \par Algorithm
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* The Gauss-Jordan method is used to find the inverse.
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* The algorithm performs a sequence of elementary row-operations until it
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* reduces the input matrix to an identity matrix. Applying the same sequence
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* of elementary row-operations to an identity matrix yields the inverse matrix.
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* If the input matrix is singular, then the algorithm terminates and returns error status
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* <code>ARM_MATH_SINGULAR</code>.
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* \image html MatrixInverse.gif "Matrix Inverse of a 3 x 3 matrix using Gauss-Jordan Method"
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*/
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/**
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* @addtogroup MatrixInv
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* @{
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*/
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/**
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* @brief Floating-point matrix inverse.
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* @param[in] *pSrc points to input matrix structure
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* @param[out] *pDst points to output matrix structure
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* @return The function returns
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* <code>ARM_MATH_SIZE_MISMATCH</code> if the input matrix is not square or if the size
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* of the output matrix does not match the size of the input matrix.
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* If the input matrix is found to be singular (non-invertible), then the function returns
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* <code>ARM_MATH_SINGULAR</code>. Otherwise, the function returns <code>ARM_MATH_SUCCESS</code>.
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*/
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arm_status arm_mat_inverse_f32(
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const arm_matrix_instance_f32 * pSrc,
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arm_matrix_instance_f32 * pDst)
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{
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float32_t *pIn = pSrc->pData; /* input data matrix pointer */
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float32_t *pOut = pDst->pData; /* output data matrix pointer */
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float32_t *pInT1, *pInT2; /* Temporary input data matrix pointer */
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float32_t *pOutT1, *pOutT2; /* Temporary output data matrix pointer */
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float32_t *pPivotRowIn, *pPRT_in, *pPivotRowDst, *pPRT_pDst; /* Temporary input and output data matrix pointer */
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uint32_t numRows = pSrc->numRows; /* Number of rows in the matrix */
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uint32_t numCols = pSrc->numCols; /* Number of Cols in the matrix */
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#if defined (ARM_MATH_DSP)
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float32_t maxC; /* maximum value in the column */
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/* Run the below code for Cortex-M4 and Cortex-M3 */
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float32_t Xchg, in = 0.0f, in1; /* Temporary input values */
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uint32_t i, rowCnt, flag = 0U, j, loopCnt, k, l; /* loop counters */
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arm_status status; /* status of matrix inverse */
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#ifdef ARM_MATH_MATRIX_CHECK
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/* Check for matrix mismatch condition */
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if ((pSrc->numRows != pSrc->numCols) || (pDst->numRows != pDst->numCols)
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|| (pSrc->numRows != pDst->numRows))
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{
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/* Set status as ARM_MATH_SIZE_MISMATCH */
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status = ARM_MATH_SIZE_MISMATCH;
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}
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else
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#endif /* #ifdef ARM_MATH_MATRIX_CHECK */
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{
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/*--------------------------------------------------------------------------------------------------------------
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* Matrix Inverse can be solved using elementary row operations.
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*
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* Gauss-Jordan Method:
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*
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* 1. First combine the identity matrix and the input matrix separated by a bar to form an
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* augmented matrix as follows:
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* _ _ _ _
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* | a11 a12 | 1 0 | | X11 X12 |
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* | | | = | |
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* |_ a21 a22 | 0 1 _| |_ X21 X21 _|
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*
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* 2. In our implementation, pDst Matrix is used as identity matrix.
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*
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* 3. Begin with the first row. Let i = 1.
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*
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* 4. Check to see if the pivot for column i is the greatest of the column.
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* The pivot is the element of the main diagonal that is on the current row.
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* For instance, if working with row i, then the pivot element is aii.
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* If the pivot is not the most significant of the columns, exchange that row with a row
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* below it that does contain the most significant value in column i. If the most
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* significant value of the column is zero, then an inverse to that matrix does not exist.
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* The most significant value of the column is the absolute maximum.
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*
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* 5. Divide every element of row i by the pivot.
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*
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* 6. For every row below and row i, replace that row with the sum of that row and
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* a multiple of row i so that each new element in column i below row i is zero.
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*
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* 7. Move to the next row and column and repeat steps 2 through 5 until you have zeros
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* for every element below and above the main diagonal.
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*
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* 8. Now an identical matrix is formed to the left of the bar(input matrix, pSrc).
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* Therefore, the matrix to the right of the bar is our solution(pDst matrix, pDst).
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*----------------------------------------------------------------------------------------------------------------*/
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/* Working pointer for destination matrix */
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pOutT1 = pOut;
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/* Loop over the number of rows */
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rowCnt = numRows;
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/* Making the destination matrix as identity matrix */
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while (rowCnt > 0U)
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{
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/* Writing all zeroes in lower triangle of the destination matrix */
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j = numRows - rowCnt;
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while (j > 0U)
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{
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*pOutT1++ = 0.0f;
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j--;
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}
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/* Writing all ones in the diagonal of the destination matrix */
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*pOutT1++ = 1.0f;
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/* Writing all zeroes in upper triangle of the destination matrix */
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j = rowCnt - 1U;
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while (j > 0U)
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{
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*pOutT1++ = 0.0f;
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j--;
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}
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/* Decrement the loop counter */
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rowCnt--;
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}
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/* Loop over the number of columns of the input matrix.
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All the elements in each column are processed by the row operations */
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loopCnt = numCols;
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/* Index modifier to navigate through the columns */
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l = 0U;
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while (loopCnt > 0U)
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{
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/* Check if the pivot element is zero..
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* If it is zero then interchange the row with non zero row below.
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* If there is no non zero element to replace in the rows below,
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* then the matrix is Singular. */
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/* Working pointer for the input matrix that points
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* to the pivot element of the particular row */
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pInT1 = pIn + (l * numCols);
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/* Working pointer for the destination matrix that points
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* to the pivot element of the particular row */
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pOutT1 = pOut + (l * numCols);
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/* Temporary variable to hold the pivot value */
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in = *pInT1;
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/* Grab the most significant value from column l */
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maxC = 0;
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for (i = l; i < numRows; i++)
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{
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maxC = *pInT1 > 0 ? (*pInT1 > maxC ? *pInT1 : maxC) : (-*pInT1 > maxC ? -*pInT1 : maxC);
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pInT1 += numCols;
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}
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/* Update the status if the matrix is singular */
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if (maxC == 0.0f)
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{
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return ARM_MATH_SINGULAR;
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}
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/* Restore pInT1 */
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pInT1 = pIn;
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/* Destination pointer modifier */
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k = 1U;
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/* Check if the pivot element is the most significant of the column */
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if ( (in > 0.0f ? in : -in) != maxC)
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{
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/* Loop over the number rows present below */
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i = numRows - (l + 1U);
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while (i > 0U)
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{
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/* Update the input and destination pointers */
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pInT2 = pInT1 + (numCols * l);
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pOutT2 = pOutT1 + (numCols * k);
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/* Look for the most significant element to
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* replace in the rows below */
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if ((*pInT2 > 0.0f ? *pInT2: -*pInT2) == maxC)
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{
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/* Loop over number of columns
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* to the right of the pilot element */
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j = numCols - l;
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while (j > 0U)
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{
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/* Exchange the row elements of the input matrix */
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Xchg = *pInT2;
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*pInT2++ = *pInT1;
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*pInT1++ = Xchg;
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/* Decrement the loop counter */
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j--;
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}
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/* Loop over number of columns of the destination matrix */
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j = numCols;
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while (j > 0U)
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{
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/* Exchange the row elements of the destination matrix */
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Xchg = *pOutT2;
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*pOutT2++ = *pOutT1;
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*pOutT1++ = Xchg;
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/* Decrement the loop counter */
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j--;
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}
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/* Flag to indicate whether exchange is done or not */
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flag = 1U;
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/* Break after exchange is done */
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break;
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}
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/* Update the destination pointer modifier */
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k++;
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/* Decrement the loop counter */
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i--;
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}
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}
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/* Update the status if the matrix is singular */
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if ((flag != 1U) && (in == 0.0f))
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{
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return ARM_MATH_SINGULAR;
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}
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/* Points to the pivot row of input and destination matrices */
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pPivotRowIn = pIn + (l * numCols);
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pPivotRowDst = pOut + (l * numCols);
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/* Temporary pointers to the pivot row pointers */
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pInT1 = pPivotRowIn;
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pInT2 = pPivotRowDst;
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/* Pivot element of the row */
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in = *pPivotRowIn;
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/* Loop over number of columns
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* to the right of the pilot element */
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j = (numCols - l);
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while (j > 0U)
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{
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/* Divide each element of the row of the input matrix
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* by the pivot element */
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in1 = *pInT1;
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*pInT1++ = in1 / in;
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/* Decrement the loop counter */
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j--;
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}
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/* Loop over number of columns of the destination matrix */
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j = numCols;
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while (j > 0U)
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{
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/* Divide each element of the row of the destination matrix
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* by the pivot element */
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in1 = *pInT2;
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*pInT2++ = in1 / in;
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/* Decrement the loop counter */
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j--;
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}
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/* Replace the rows with the sum of that row and a multiple of row i
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* so that each new element in column i above row i is zero.*/
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/* Temporary pointers for input and destination matrices */
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pInT1 = pIn;
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pInT2 = pOut;
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/* index used to check for pivot element */
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i = 0U;
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/* Loop over number of rows */
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/* to be replaced by the sum of that row and a multiple of row i */
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k = numRows;
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while (k > 0U)
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{
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/* Check for the pivot element */
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if (i == l)
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{
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/* If the processing element is the pivot element,
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only the columns to the right are to be processed */
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pInT1 += numCols - l;
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pInT2 += numCols;
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}
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else
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{
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/* Element of the reference row */
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in = *pInT1;
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/* Working pointers for input and destination pivot rows */
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pPRT_in = pPivotRowIn;
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pPRT_pDst = pPivotRowDst;
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/* Loop over the number of columns to the right of the pivot element,
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to replace the elements in the input matrix */
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j = (numCols - l);
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while (j > 0U)
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{
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/* Replace the element by the sum of that row
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and a multiple of the reference row */
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in1 = *pInT1;
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*pInT1++ = in1 - (in * *pPRT_in++);
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/* Decrement the loop counter */
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j--;
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}
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/* Loop over the number of columns to
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replace the elements in the destination matrix */
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j = numCols;
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while (j > 0U)
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{
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/* Replace the element by the sum of that row
|
||
|
|
and a multiple of the reference row */
|
||
|
|
in1 = *pInT2;
|
||
|
|
*pInT2++ = in1 - (in * *pPRT_pDst++);
|
||
|
|
|
||
|
|
/* Decrement the loop counter */
|
||
|
|
j--;
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Increment the temporary input pointer */
|
||
|
|
pInT1 = pInT1 + l;
|
||
|
|
|
||
|
|
/* Decrement the loop counter */
|
||
|
|
k--;
|
||
|
|
|
||
|
|
/* Increment the pivot index */
|
||
|
|
i++;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Increment the input pointer */
|
||
|
|
pIn++;
|
||
|
|
|
||
|
|
/* Decrement the loop counter */
|
||
|
|
loopCnt--;
|
||
|
|
|
||
|
|
/* Increment the index modifier */
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
#else
|
||
|
|
|
||
|
|
/* Run the below code for Cortex-M0 */
|
||
|
|
|
||
|
|
float32_t Xchg, in = 0.0f; /* Temporary input values */
|
||
|
|
uint32_t i, rowCnt, flag = 0U, j, loopCnt, k, l; /* loop counters */
|
||
|
|
arm_status status; /* status of matrix inverse */
|
||
|
|
|
||
|
|
#ifdef ARM_MATH_MATRIX_CHECK
|
||
|
|
|
||
|
|
/* Check for matrix mismatch condition */
|
||
|
|
if ((pSrc->numRows != pSrc->numCols) || (pDst->numRows != pDst->numCols)
|
||
|
|
|| (pSrc->numRows != pDst->numRows))
|
||
|
|
{
|
||
|
|
/* Set status as ARM_MATH_SIZE_MISMATCH */
|
||
|
|
status = ARM_MATH_SIZE_MISMATCH;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
#endif /* #ifdef ARM_MATH_MATRIX_CHECK */
|
||
|
|
{
|
||
|
|
|
||
|
|
/*--------------------------------------------------------------------------------------------------------------
|
||
|
|
* Matrix Inverse can be solved using elementary row operations.
|
||
|
|
*
|
||
|
|
* Gauss-Jordan Method:
|
||
|
|
*
|
||
|
|
* 1. First combine the identity matrix and the input matrix separated by a bar to form an
|
||
|
|
* augmented matrix as follows:
|
||
|
|
* _ _ _ _ _ _ _ _
|
||
|
|
* | | a11 a12 | | | 1 0 | | | X11 X12 |
|
||
|
|
* | | | | | | | = | |
|
||
|
|
* |_ |_ a21 a22 _| | |_0 1 _| _| |_ X21 X21 _|
|
||
|
|
*
|
||
|
|
* 2. In our implementation, pDst Matrix is used as identity matrix.
|
||
|
|
*
|
||
|
|
* 3. Begin with the first row. Let i = 1.
|
||
|
|
*
|
||
|
|
* 4. Check to see if the pivot for row i is zero.
|
||
|
|
* The pivot is the element of the main diagonal that is on the current row.
|
||
|
|
* For instance, if working with row i, then the pivot element is aii.
|
||
|
|
* If the pivot is zero, exchange that row with a row below it that does not
|
||
|
|
* contain a zero in column i. If this is not possible, then an inverse
|
||
|
|
* to that matrix does not exist.
|
||
|
|
*
|
||
|
|
* 5. Divide every element of row i by the pivot.
|
||
|
|
*
|
||
|
|
* 6. For every row below and row i, replace that row with the sum of that row and
|
||
|
|
* a multiple of row i so that each new element in column i below row i is zero.
|
||
|
|
*
|
||
|
|
* 7. Move to the next row and column and repeat steps 2 through 5 until you have zeros
|
||
|
|
* for every element below and above the main diagonal.
|
||
|
|
*
|
||
|
|
* 8. Now an identical matrix is formed to the left of the bar(input matrix, src).
|
||
|
|
* Therefore, the matrix to the right of the bar is our solution(dst matrix, dst).
|
||
|
|
*----------------------------------------------------------------------------------------------------------------*/
|
||
|
|
|
||
|
|
/* Working pointer for destination matrix */
|
||
|
|
pOutT1 = pOut;
|
||
|
|
|
||
|
|
/* Loop over the number of rows */
|
||
|
|
rowCnt = numRows;
|
||
|
|
|
||
|
|
/* Making the destination matrix as identity matrix */
|
||
|
|
while (rowCnt > 0U)
|
||
|
|
{
|
||
|
|
/* Writing all zeroes in lower triangle of the destination matrix */
|
||
|
|
j = numRows - rowCnt;
|
||
|
|
while (j > 0U)
|
||
|
|
{
|
||
|
|
*pOutT1++ = 0.0f;
|
||
|
|
j--;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Writing all ones in the diagonal of the destination matrix */
|
||
|
|
*pOutT1++ = 1.0f;
|
||
|
|
|
||
|
|
/* Writing all zeroes in upper triangle of the destination matrix */
|
||
|
|
j = rowCnt - 1U;
|
||
|
|
while (j > 0U)
|
||
|
|
{
|
||
|
|
*pOutT1++ = 0.0f;
|
||
|
|
j--;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Decrement the loop counter */
|
||
|
|
rowCnt--;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Loop over the number of columns of the input matrix.
|
||
|
|
All the elements in each column are processed by the row operations */
|
||
|
|
loopCnt = numCols;
|
||
|
|
|
||
|
|
/* Index modifier to navigate through the columns */
|
||
|
|
l = 0U;
|
||
|
|
//for(loopCnt = 0U; loopCnt < numCols; loopCnt++)
|
||
|
|
while (loopCnt > 0U)
|
||
|
|
{
|
||
|
|
/* Check if the pivot element is zero..
|
||
|
|
* If it is zero then interchange the row with non zero row below.
|
||
|
|
* If there is no non zero element to replace in the rows below,
|
||
|
|
* then the matrix is Singular. */
|
||
|
|
|
||
|
|
/* Working pointer for the input matrix that points
|
||
|
|
* to the pivot element of the particular row */
|
||
|
|
pInT1 = pIn + (l * numCols);
|
||
|
|
|
||
|
|
/* Working pointer for the destination matrix that points
|
||
|
|
* to the pivot element of the particular row */
|
||
|
|
pOutT1 = pOut + (l * numCols);
|
||
|
|
|
||
|
|
/* Temporary variable to hold the pivot value */
|
||
|
|
in = *pInT1;
|
||
|
|
|
||
|
|
/* Destination pointer modifier */
|
||
|
|
k = 1U;
|
||
|
|
|
||
|
|
/* Check if the pivot element is zero */
|
||
|
|
if (*pInT1 == 0.0f)
|
||
|
|
{
|
||
|
|
/* Loop over the number rows present below */
|
||
|
|
for (i = (l + 1U); i < numRows; i++)
|
||
|
|
{
|
||
|
|
/* Update the input and destination pointers */
|
||
|
|
pInT2 = pInT1 + (numCols * l);
|
||
|
|
pOutT2 = pOutT1 + (numCols * k);
|
||
|
|
|
||
|
|
/* Check if there is a non zero pivot element to
|
||
|
|
* replace in the rows below */
|
||
|
|
if (*pInT2 != 0.0f)
|
||
|
|
{
|
||
|
|
/* Loop over number of columns
|
||
|
|
* to the right of the pilot element */
|
||
|
|
for (j = 0U; j < (numCols - l); j++)
|
||
|
|
{
|
||
|
|
/* Exchange the row elements of the input matrix */
|
||
|
|
Xchg = *pInT2;
|
||
|
|
*pInT2++ = *pInT1;
|
||
|
|
*pInT1++ = Xchg;
|
||
|
|
}
|
||
|
|
|
||
|
|
for (j = 0U; j < numCols; j++)
|
||
|
|
{
|
||
|
|
Xchg = *pOutT2;
|
||
|
|
*pOutT2++ = *pOutT1;
|
||
|
|
*pOutT1++ = Xchg;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Flag to indicate whether exchange is done or not */
|
||
|
|
flag = 1U;
|
||
|
|
|
||
|
|
/* Break after exchange is done */
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Update the destination pointer modifier */
|
||
|
|
k++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Update the status if the matrix is singular */
|
||
|
|
if ((flag != 1U) && (in == 0.0f))
|
||
|
|
{
|
||
|
|
return ARM_MATH_SINGULAR;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Points to the pivot row of input and destination matrices */
|
||
|
|
pPivotRowIn = pIn + (l * numCols);
|
||
|
|
pPivotRowDst = pOut + (l * numCols);
|
||
|
|
|
||
|
|
/* Temporary pointers to the pivot row pointers */
|
||
|
|
pInT1 = pPivotRowIn;
|
||
|
|
pOutT1 = pPivotRowDst;
|
||
|
|
|
||
|
|
/* Pivot element of the row */
|
||
|
|
in = *(pIn + (l * numCols));
|
||
|
|
|
||
|
|
/* Loop over number of columns
|
||
|
|
* to the right of the pilot element */
|
||
|
|
for (j = 0U; j < (numCols - l); j++)
|
||
|
|
{
|
||
|
|
/* Divide each element of the row of the input matrix
|
||
|
|
* by the pivot element */
|
||
|
|
*pInT1 = *pInT1 / in;
|
||
|
|
pInT1++;
|
||
|
|
}
|
||
|
|
for (j = 0U; j < numCols; j++)
|
||
|
|
{
|
||
|
|
/* Divide each element of the row of the destination matrix
|
||
|
|
* by the pivot element */
|
||
|
|
*pOutT1 = *pOutT1 / in;
|
||
|
|
pOutT1++;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Replace the rows with the sum of that row and a multiple of row i
|
||
|
|
* so that each new element in column i above row i is zero.*/
|
||
|
|
|
||
|
|
/* Temporary pointers for input and destination matrices */
|
||
|
|
pInT1 = pIn;
|
||
|
|
pOutT1 = pOut;
|
||
|
|
|
||
|
|
for (i = 0U; i < numRows; i++)
|
||
|
|
{
|
||
|
|
/* Check for the pivot element */
|
||
|
|
if (i == l)
|
||
|
|
{
|
||
|
|
/* If the processing element is the pivot element,
|
||
|
|
only the columns to the right are to be processed */
|
||
|
|
pInT1 += numCols - l;
|
||
|
|
pOutT1 += numCols;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
/* Element of the reference row */
|
||
|
|
in = *pInT1;
|
||
|
|
|
||
|
|
/* Working pointers for input and destination pivot rows */
|
||
|
|
pPRT_in = pPivotRowIn;
|
||
|
|
pPRT_pDst = pPivotRowDst;
|
||
|
|
|
||
|
|
/* Loop over the number of columns to the right of the pivot element,
|
||
|
|
to replace the elements in the input matrix */
|
||
|
|
for (j = 0U; j < (numCols - l); j++)
|
||
|
|
{
|
||
|
|
/* Replace the element by the sum of that row
|
||
|
|
and a multiple of the reference row */
|
||
|
|
*pInT1 = *pInT1 - (in * *pPRT_in++);
|
||
|
|
pInT1++;
|
||
|
|
}
|
||
|
|
/* Loop over the number of columns to
|
||
|
|
replace the elements in the destination matrix */
|
||
|
|
for (j = 0U; j < numCols; j++)
|
||
|
|
{
|
||
|
|
/* Replace the element by the sum of that row
|
||
|
|
and a multiple of the reference row */
|
||
|
|
*pOutT1 = *pOutT1 - (in * *pPRT_pDst++);
|
||
|
|
pOutT1++;
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
/* Increment the temporary input pointer */
|
||
|
|
pInT1 = pInT1 + l;
|
||
|
|
}
|
||
|
|
/* Increment the input pointer */
|
||
|
|
pIn++;
|
||
|
|
|
||
|
|
/* Decrement the loop counter */
|
||
|
|
loopCnt--;
|
||
|
|
/* Increment the index modifier */
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
#endif /* #if defined (ARM_MATH_DSP) */
|
||
|
|
|
||
|
|
/* Set status as ARM_MATH_SUCCESS */
|
||
|
|
status = ARM_MATH_SUCCESS;
|
||
|
|
|
||
|
|
if ((flag != 1U) && (in == 0.0f))
|
||
|
|
{
|
||
|
|
pIn = pSrc->pData;
|
||
|
|
for (i = 0; i < numRows * numCols; i++)
|
||
|
|
{
|
||
|
|
if (pIn[i] != 0.0f)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
|
||
|
|
if (i == numRows * numCols)
|
||
|
|
status = ARM_MATH_SINGULAR;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
/* Return to application */
|
||
|
|
return (status);
|
||
|
|
}
|
||
|
|
|
||
|
|
/**
|
||
|
|
* @} end of MatrixInv group
|
||
|
|
*/
|