See Also LinearAlgebra/Details LinearAlgebra[Modular] LinearAlgebra[Modular][AddMultiple] LinearAlgebra[Modular][Copy] LinearAlgebra[Modular][Create] LinearAlgebra[Modular][Determinant] LinearAlgebra[Modular][Mod] LinearAlgebra[Modular][Multiply]

Specifying incrow as a name accomplishes two things. Firstly, it indicates that the algorithm continues computation even if inconsistent rows are found, placing those rows immediately after the consistent rows in the Matrix upon completion. Secondly, it indicates that the algorithm assigns the number of inconsistent rows found to incrow upon completion. If you specify incrow=0, the algorithm halts with an error when inconsistent rows are found.

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The signature of a Matrix is an integer checksum that represents the correspondence of the solving columns and corresponding rows in the row echelon or reduced row echelon form of a Matrix. If for two Matrices, the pattern of pivots (location of the leading 1 entries in each row) is the same, their signature is also the same. However, two signatures being equal does not necessarily mean the two Matrices have the same pattern of pivots. Use this value is rapidly detect bad homomorphisms for reconstruction of the solution of an integer linear system.

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The RowReduce function performs in-place Gauss elimination on the input mod p Matrix A. If redflag=false, it produces a row echelon form. If redflag=true, it produces a reduced row echelon form.

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RowReduce can act on a sub-Matrix or augmented Matrix based on the values of rows, cols, and rcol. The parameters rows and cols can be integer values, or nonempty ranges with integer endpoints, which represent the total rows and columns (including any augmented columns) on which the row reduction should be performed.  If specified as integer values, the range is assumed to start at 1.

We also offer custom gear manufacturing. We can manufacture gears in various materials such as plastic, brass, steel and aluminum. We can manufacture spur gears, bevel gears, helical gears, gear clusters, metric gears, anti-backlash gears, worm gears, gear blanks, gear stock, pinion shafts, ratchets and pawls. We can also manufacture gears with pin hubs, clamp hubs, or hubless.

Generally, it is required that p be a prime, as inverses are needed, but in some cases it is possible to obtain row echelon form when p is composite. For the cases where row echelon form cannot be obtained for p composite, the function returns an error indicating that the algorithm failed because p is composite.

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B&B Manufacturing® is a trusted supplier of quality gears for original equipment manufacturers (OEM), distributors, and the power transmission and precision mechanical markets, as well as the government and military.

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B&B Manufacturing® offers external spur gears from stock in brass, steel, stainless steel, and cast iron. Our stock spur gears are offered with hubs and an overall length of 0.437” through the bore. Our stock spur gears have a pressure angle of either 14.5 or 20 degrees, a variety of minimum plain bore and finished bore options, and a variety of diametric. All B&B Manufacturing® gears meet and exceed AGMA quality standards and our own precision manufacturing tolerances.

Taper-Lock®, HTD®, Poly Chain®, PowerGrip®, GT2®, and GT3® are trademarks of the Gates® Corporation. QD® is a trademark of TB Woods, Inc. FHT® is a trademark of Fenner Precision.

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If the pseudo-determinant of the Matrix being reduced is required, specify pdet as a name. Otherwise, specify pdet as the value 0. On output, the name specified for pdet is assigned the value of the pseudo-determinant. If the input Matrix has fewer variable columns than rows, an error is produced.

p - modulus A - mod p Matrix rows - number of rows, or range of rows to use cols - number of columns, or range of columns to use rcol - number of columns that represent variables det - name to use for output determinant; 0 to disable pdet - name to use for output pseudo-determinant; 0 to disable rank - name to use for output rank; 0 to disable sig - name to use for output signature; 0 to disable incrow - name to use for output inconsistent row count; 0 to disable redflag - boolean or integer; indicate whether to reduce row echelon form, or for how many rows

For a standard n⁢x⁢m Matrix, augmented with two columns, specify rows=n, cols=m+2, and rcol=m. For a partially reduced n⁢x⁢n Matrix with one augmented column, where the first r rows have leading identity, the reduction of the remaining rows can be accomplished by specifying rows=r..n, cols=r..n+1, and rcol=n−r+1.

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If the determinant of the Matrix being reduced is required, specify det as a name. Otherwise, specify det as the value 0. On output, the name specified for det is assigned the value of the determinant. If the input Matrix has more variable columns than rows, the determinant is computed from the square Matrix formed by removing the extra columns from the end of the Matrix. If the input Matrix has fewer variable columns than rows, an error is produced.

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form RowReduce(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][RowReduce](..).

Calling Sequence RowReduce(p, A, rows, cols, rcol, det, pdet, rank, sig, incrow, redflag) Parameters p - modulus A - mod p Matrix rows - number of rows, or range of rows to use cols - number of columns, or range of columns to use rcol - number of columns that represent variables det - name to use for output determinant; 0 to disable pdet - name to use for output pseudo-determinant; 0 to disable rank - name to use for output rank; 0 to disable sig - name to use for output signature; 0 to disable incrow - name to use for output inconsistent row count; 0 to disable redflag - boolean or integer; indicate whether to reduce row echelon form, or for how many rows Description •  The RowReduce function performs in-place Gauss elimination on the input mod p Matrix A. If redflag=false, it produces a row echelon form. If redflag=true, it produces a reduced row echelon form. •  Generally, it is required that p be a prime, as inverses are needed, but in some cases it is possible to obtain row echelon form when p is composite. For the cases where row echelon form cannot be obtained for p composite, the function returns an error indicating that the algorithm failed because p is composite. •  RowReduce can act on a sub-Matrix or augmented Matrix based on the values of rows, cols, and rcol. The parameters rows and cols can be integer values, or nonempty ranges with integer endpoints, which represent the total rows and columns (including any augmented columns) on which the row reduction should be performed.  If specified as integer values, the range is assumed to start at 1.    For a standard n⁢x⁢m Matrix, augmented with two columns, specify rows=n, cols=m+2, and rcol=m. For a partially reduced n⁢x⁢n Matrix with one augmented column, where the first r rows have leading identity, the reduction of the remaining rows can be accomplished by specifying rows=r..n, cols=r..n+1, and rcol=n−r+1. •  If the determinant of the Matrix being reduced is required, specify det as a name. Otherwise, specify det as the value 0. On output, the name specified for det is assigned the value of the determinant. If the input Matrix has more variable columns than rows, the determinant is computed from the square Matrix formed by removing the extra columns from the end of the Matrix. If the input Matrix has fewer variable columns than rows, an error is produced.    Note: If the determinant is not required, set det to zero to avoid unnecessary calculations. •  The pseudo-determinant of a Matrix is the determinant of the Matrix formed by removing any dependent rows and parametric columns (up to sign). In cases where the input ( n⁢x⁢m ) Matrix is of full rank with respect to the first n⁢x⁢n submatrix, the determinant and pseudo-determinant are the same. Use this value for reconstruction of solutions of integer linear systems that are rank deficient or underdetermined.    If the pseudo-determinant of the Matrix being reduced is required, specify pdet as a name. Otherwise, specify pdet as the value 0. On output, the name specified for pdet is assigned the value of the pseudo-determinant. If the input Matrix has fewer variable columns than rows, an error is produced.    Note: If the pseudo-determinant is not required, set pdet to zero to avoid unnecessary calculations. •  The signature of a Matrix is an integer checksum that represents the correspondence of the solving columns and corresponding rows in the row echelon or reduced row echelon form of a Matrix. If for two Matrices, the pattern of pivots (location of the leading 1 entries in each row) is the same, their signature is also the same. However, two signatures being equal does not necessarily mean the two Matrices have the same pattern of pivots. Use this value is rapidly detect bad homomorphisms for reconstruction of the solution of an integer linear system.    If the signature of the Matrix being reduced is required, specify sig as a name. Otherwise, specify sig as the value 0. On output, the name specified for sig is assigned the value of the signature. •  If the rank of the Matrix being reduced is required, specify rank as a name. Otherwise, specify rank as the value 0. On output, the name specified for rank is assigned the rank of the Matrix. •  Specifying incrow as a name accomplishes two things. Firstly, it indicates that the algorithm continues computation even if inconsistent rows are found, placing those rows immediately after the consistent rows in the Matrix upon completion. Secondly, it indicates that the algorithm assigns the number of inconsistent rows found to incrow upon completion. If you specify incrow=0, the algorithm halts with an error when inconsistent rows are found. •  The redflag parameter controls whether the output should be a reduced row echelon form (true) or a standard row echelon form with no back-substitution (false). In addition to these all-or-nothing settings, one can specify that the first i rows of a matrix should have back-substitution performed on them if redflag is set to −i. •  This command is part of the LinearAlgebra[Modular] package, so it can be used in the form RowReduce(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][RowReduce](..). Examples For this basic augmented system, find the rank and determinant. >  with⁡LinearAlgebraModular: >  p≔2741 p≔2741 (1) >  A≔Mod⁡p,Matrix⁡4,5,i,j↦rand⁡,integer A≔25431568127356581430154923761511183916419462114924183014807541049423 (2) >  B≔Copy⁡p,A: >  RowReduce⁡p,B,4,5,4,det,0,rank,0,0,true: >  B,rank 1000191601006590010181000125,4 (3) Check the determinant. >  det 82 (4) >  modp⁡LinearAlgebra:-Determinant⁡Matrix⁡A1..4,1..4,datatype=integer,p 82 (5) Check the solution. >  Multiply⁡p,A,1..4,1..4,B,1..4,5,Copy⁡p,A,1..4,5 58118392418423,58118392418423 (6) Consider the system solution. Two steps looking at submatrices. >  A≔Mod⁡p,Matrix⁡4,5,i,j↦rand⁡,float8 A≔1980.2533.1439.67.2051.2635.353.2657.1617.2198.2587.1857.827.1848.2338.1720.1181.493.1731.2264. (7) >  B≔Copy⁡p,A: First step, working with the first two rows of B. >  RowReduce⁡p,B,1..2,5,4,det1,0,0,0,0,true: >  B 1.0.732.2653.1849.0.1.1097.941.1501.2587.1857.827.1848.2338.1720.1181.493.1731.2264. (8) Forward substitute. >  forito2doforjto2doAddMultiple⁡p,p−B2+i,j,B,2+i,B,j,B,2+ienddoenddo:B 1.0.732.2653.1849.0.1.1097.941.1501.0.0.608.581.2260.0.0.508.1120.2290. (9) Work with the remaining two rows of B. >  RowReduce⁡p,B,3..4,3..5,2,det2,0,0,0,0,true: >  B 1.0.732.2653.1849.0.1.1097.941.1501.0.0.1.0.2413.0.0.0.1.1536. (10) Back-substitute last two rows with entries in first two by using shortcut specification for last three columns. >  forito2doforjto2doAddMultiple⁡p,p−Bi,2+j,B,i,3..5,B,j+2,3,B,i,3enddoenddo:B 1.0.0.0.1596.0.1.0.0.1377.0.0.1.0.2413.0.0.0.1.1536. (11) Check the determinant. >  modp⁡round⁡det1⁢det2,p 2603 (12) >  modp⁡LinearAlgebra:-Determinant⁡Matrix⁡4,4,i,j↦round⁡Ai,j,datatype=integer,p 2603 (13) Check the solution. >  Multiply⁡p,A,1..4,1..4,B,1..4,5,Copy⁡p,A,1..4,5 2051.2198.2338.2264.,2051.2198.2338.2264. (14) Back substitution only acting on the first row >  p≔65521 p≔65521 (15) >  M≔Create⁡p,4,5,random,integer M≔376066440307911786645834553816159723313465421456079636696651356371352874493385050550237621146483 (16) >  RowReduce⁡p,M,4,5,4,0,0,0,0,0,−1: >  M 100017540014564514195369120016472852075000141141 (17) See Also LinearAlgebra/Details LinearAlgebra[Modular] LinearAlgebra[Modular][AddMultiple] LinearAlgebra[Modular][Copy] LinearAlgebra[Modular][Create] LinearAlgebra[Modular][Determinant] LinearAlgebra[Modular][Mod] LinearAlgebra[Modular][Multiply]

Available in metric and U.S. (inch) sizes, our gears meet AGMA (American Gear Manufacturers Association) standards, feature customized numbers of teeth, and are offered in both 14.5 degree pressure angle and 20 degree pressure angle. The following diametrical inch pitches are available:

B&B Manufacturing will be closed December 24th & 25th for the holidays and December 31st & January 1st for the new year.

If the signature of the Matrix being reduced is required, specify sig as a name. Otherwise, specify sig as the value 0. On output, the name specified for sig is assigned the value of the signature.

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Taper-Lock®, HTD®, Poly Chain®, PowerGrip®, GT2®, and GT3® are trademarks of the Gates® Corporation. QD® is a trademark of TB Woods, Inc. FHT® is a trademark of Fenner Precision.

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What is a spur gear? A spur gear is a straight cut gear with the gear hobbing on the rim of the gear and parallel to the bore and rotating axis of the gear. Spur gears are one of the most common and cost-effective types of precision gears. Spur gears mechanically transmit motion and power between parallel shafts, which rotate in opposite directions.

forito2doforjto2doAddMultiple⁡p,p−Bi,2+j,B,i,3..5,B,j+2,3,B,i,3enddoenddo:B

If the rank of the Matrix being reduced is required, specify rank as a name. Otherwise, specify rank as the value 0. On output, the name specified for rank is assigned the rank of the Matrix.

The pseudo-determinant of a Matrix is the determinant of the Matrix formed by removing any dependent rows and parametric columns (up to sign). In cases where the input ( n⁢x⁢m ) Matrix is of full rank with respect to the first n⁢x⁢n submatrix, the determinant and pseudo-determinant are the same. Use this value for reconstruction of solutions of integer linear systems that are rank deficient or underdetermined.

The redflag parameter controls whether the output should be a reduced row echelon form (true) or a standard row echelon form with no back-substitution (false). In addition to these all-or-nothing settings, one can specify that the first i rows of a matrix should have back-substitution performed on them if redflag is set to −i.

Egg beaters, Electric screwdrivers, mechanical lamps and lighting, oscillating sprinklers, windup alarm clocks, clothes washers and dryers, conveyors, and blenders are a few of the many machines using spur gears. Spur gears are also used in winches, construction equipment, outboard motors, machine tools, lawn and garden equipment, marine hoists, multi-spindle drives, snow throwers, turbine drives, pumps, centrifuges, medical devices, jacks and other power transmission equipment.