Apparatus and method for inverting a 4×4 matrix

An article comprising a machine readable medium that stores data representing a predetermined function, the predetermined function comprising:

• dividing a source matrix into four 2×2 sub-matrices, A, B, C and D;
• calculating one or more intermediate sub-matrix products from one or more of the sub-matrices;
• calculating a determinant of the source matrix to form a determinant residue rd utilizing the intermediate sub-matrix products;
• scaling a determinant of each sub-matrix and the intermediate sub-matrix products using determinant residue rd to form final sub-matrix products;
• forming a partial inverse sub-matrix pA, pB, pC and pD for each sub-matrix using the scaled sub-matrix determinants and the final sub-matrix products;
• and
• calculating an inverse of each sub-matrix iA, iB, iC and iD, utilizing each partial inverse sub-matrix to form an inverse source matrix iS.

A computer readable storage medium including program instructions that direct a computer to function in a specified manner when executed by a processor, the program instructions comprising:

• dividing the source matrix into four 2×2 sub-matrices A, B, C and D;
• calculating a plurality of sub-matrix products from the sub-matrices;
• calculating a determinant of the source matrix dS to form a matrix determinant residue rd of the source matrix as rd=1/dS;
• forming a partial, inverse sub-matrix of each sub-matrix using one or more of the matrix products and a determinant of each sub-matrix;
• and
• calculating an inverse of each sub-matrix iA, iB, iC, and iD, utilizing each partial, inverse sub-matrix and determinant residue rd, such that an inverse of the source matrix iS is formed.

The computer readable storage medium including program instructions that direct a computer to function in a specified manner when executed by a processor, the program instructions comprising:

• dividing a source matrix into four 2×2 sub-matrices, A, B, C and D;
• calculating one or more intermediate sub-matrix products from one or more of the sub-matrices;
• calculating a determinant of the source matrix dS to form a determinant residue rd of the source matrix utilizing the intermediate sub-matrix products and the sub-matrix determinants;
• scaling a determinant of each sub-matrix and the intermediate sub-matrix products using determinant residue rd to form final sub-matrix products;
• forming a partial inverse sub-matrix pA, pB, pC and pD for each sub-matrix using the scaled sub-matrix determinants and the final sub-matrix products;
• and
• calculating an inverse of each sub-matrix iA, iB, iC and iD, utilizing each partial inverse sub-matrix to form an inverse source matrix iS.

An apparatus, comprising:

• a processor having circuitry to execute instructions;
• a plurality of SIMD data storage devices coupled to the processor, the SIMD data storage registers to pairs of floating point vectors during matrix calculation;
• a storage device coupled to the processor, having sequences of instructions stored therein, which when executed by the processor cause the processor to:
• divide the source matrix into four 2×2 sub-matrices A, B, C and D;
• calculate a plurality of sub-matrix products from the sub-matrices;
• calculate a determinant of the source matrix dS to form a determinant residue rd of the source matrix as rd=1/dS;
• form a partial, inverse sub-matrix of each sub-matrix using one or more of the matrix products and the determinant of each sub-matrix;
• and
• calculate an inverse of each sub-matrix iA, iB, iC, and iD, utilizing each partial, inverse sub-matrix and determinant residue rd, such that an inverse of the source matrix iS is formed.

A system, comprising:

• a processor having circuitry to execute instructions;
• a plurality of SIMD data storage devices coupled to the processor, the SIMD data storage registers to pairs of floating point vectors during matrix calculation;
• a storage device coupled to the processor, having sequences of instructions stored therein, which when executed by the processor cause the processor to:
• divide a source matrix into four 2×2 sub-matrices, A, B, C and D;
• calculate one or more intermediate sub-matrix products from each of the sub-matrices,
• calculate a source matrix dS to form a determinant residue rd utilizing the intermediate sub-matrix products,
• scale a determinant of each sub-matrix and the intermediate sub-matrix products using determinant residue rd to form final sub-matrix products,
• form a partial inverse sub-matrix pA, pB, pC and pD for each sub-matrix using the scaled sub-matrix determinants and the final sub-matrix products, and
• calculate an inverse of each sub-matrix iA, iB, iC and iD, utilizing each partial inverse sub-matrix to form an inverse source matrix iS.

A method comprising:

• dividing a source matrix into four 2×2 sub-matrices A, B, C and D;
• storing each two element row of each 2×2 sub-matrix within a single instruction multiple data (SIMD) register;
• forming a partial, inverse sub-matrix of each sub-matrix using one or more of a plurality of sub-matrix products calculated from the sub-matrices and a determinant of each sub-matrix within one or more SIMD registers;
• and
• calculating an inverse of each sub-matrix iA, iB, iC and iD, utilizing each partial, inverse sub-matrix and a determinant residue rd calculated from the source matrix, such that an inverse of the source matrix iS is formed within the one or more SIMD registers.