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RegularChains[MatrixTools][MatrixMultiply] - compute the product of two matrices modulo a regular chain
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Calling Sequence
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MatrixMultiply(A, B, rc, R)
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Parameters
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A
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Matrix with coefficients in the field of fractions of R
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B
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Matrix with coefficients in the field of fractions of R
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rc
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regular chain of R
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R
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polynomial ring
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Description
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The command MatrixMultiply(A, B, rc, R) returns the product of A and B mod the saturated ideal of rc.
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The result is viewed as a matrix with coefficients in the total ring of fractions of R/I where I is the saturated ideal of rc.
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The implementation is based on the method proposed in the paper "On {W}inograd's Algorithm for Inner Products" by A. Waksman.
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It is assumed that rc is strongly normalized.
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This command is part of the RegularChains[MatrixTools] package, so it can be used in the form MatrixMultiply(..) only after executing the command with(RegularChains[MatrixTools]). However, it can always be accessed through the long form of the command by using
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Examples
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![R := PolynomialRing([y, z])](/support/helpjp/helpview.aspx?si=4970/file06507/math118.png)
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![rc := Chain([z^4+1, y^2-z^2], rc, R)](/support/helpjp/helpview.aspx?si=4970/file06507/math132.png)
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![[y^2-z^2, z^4+1]](/support/helpjp/helpview.aspx?si=4970/file06507/math139.png)
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![m := Matrix([[1, y+z], [0, y-z]])](/support/helpjp/helpview.aspx?si=4970/file06507/math143.png)
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![m := Matrix([[1, y+z], [0, y-z]])](/support/helpjp/helpview.aspx?si=4970/file06507/math146.png)
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![mim := [[[Matrix([[1, 0], [0, (1/2)*z^3]]), regular_chain]], [["noInv", Matrix([[1, y+z], [0, y-z]]), regular_chain]]]](/support/helpjp/helpview.aspx?si=4970/file06507/math153.png)
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![m1 := mim[1][1][1]](/support/helpjp/helpview.aspx?si=4970/file06507/math157.png)
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![m1 := Matrix([[1, 0], [0, (1/2)*z^3]])](/support/helpjp/helpview.aspx?si=4970/file06507/math160.png)
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![rc1 := mim[1][1][2]](/support/helpjp/helpview.aspx?si=4970/file06507/math164.png)
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| (7) |
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![Matrix([[1, 0], [0, 1]])](/support/helpjp/helpview.aspx?si=4970/file06507/math174.png)
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See Also
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Chain, Empty, Equations, IsStronglyNormalized, IsZeroMatrix, JacobianMatrix, LowerEchelonForm, Matrix, MatrixInverse, MatrixOverChain, MatrixTools, NormalForm, PolynomialRing, RegularChains
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References
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A. Waksman "On Winograd's Algorithm for Inner Products." IEEE Transactions On Computers, C-19, (1970): 360-361.
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