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Efficiency Improvements in Maple 9.5

  

A number of efficiency improvements have been applied to Maple. Maple 9.5 is faster and uses less memory.

 

Quotients of Polynomials

Indefinite Rational Summation

Operations on Matrix Sub-blocks

Rtable Type Checks

Conversions Between diff and D

Numerical Linear Algebra

Quotients of Polynomials

  

Maple 9.5 is more efficient when computing the quotient of sparse polynomials.

Example

p1 := 25000000000000000000*x^24*(234374765625000000*(x^(1/5000000000*a))^609256031*y^9256031 + 7946285261481299*x^12*y^27768093 - 56259267500000000*(x^(1/10000000000*a))^609256031*x^6*z^9256031)^4;

p125000000000000000000x24234374765625000000xa5000000000609256031y9256031+7946285261481299x12y2776809356259267500000000xa10000000000609256031x6z92560314

(1)

p2 := y^100000000-x;

p2y100000000x

(2)

t0 := time():

res := quo(p1, p2, y);

res99677380232696642539570742968540066928124035645820992834105370025000000000000000000x72y11072372

(3)

time_consumed = time()-t0;

time_consumed=0.008

(4)

Indefinite Rational Summation

  

The algorithm of Gerhard, Giesbrecht, Storjohann, and Zima for indefinite rational summation has been incorporated into SumTools[Hypergeometric][SumDecomposition]. It is faster than the previous algorithm when the dispersion of the input is large.

Example

ee := (x^2+2000*x+1001000)/((x+1001)*(x+1000)*x);

eex2+2000x+1001000x+1001x+1000x

(5)

t0 := time():

Sum(ee,x) = sum(ee,x);

xx2+2000x+1001000x+1001x+1000x=1x+1000+Ψx

(6)

time_consumed = time() - t0;

time_consumed=0.079

(7)

Operations on Matrix Sub-blocks

• 

The new rtable_scanblock command performs efficient block-operations on regions of a Matrix, Vector, or Array.

Rtable Type Checks

  

Type checks of the form type(M,'Matrix'(algebraic)), which usually need to scan the entries of the given Matrix, Vector, or Array are much faster in Maple 9.5.

Example

M := LinearAlgebra:-RandomMatrix(500);

(8)

time_consumed = time( type(M, 'Matrix'(algebraic)) );

time_consumed=0.011

(9)

Conversions Between diff and D

  

Maple 9.5 includes faster conversions between the diff and D notations for derivatives.

Example

diff( f(x,y,z,t), x$74, y$143, z$236 );

453x74y143z236fx,y,z,t

(10)

time_consumed = time( convert( (10), D ) );  # before it was 1/2 minute in typical machines

time_consumed=0.012

(11)

Numerical Linear Algebra

Windows

  

Maple 9.5 for Windows uses version 6.1 of the Intel Math Kernel Library (MKL) resulting in faster computation for large-sized hardware floating-point linear algebra operations.

  

The following example takes approximately 6.7 seconds in Maple 9 and 4.0 seconds in Maple 9.5 on a 1.7GHz P4 machine running Windows 2000.

M := LinearAlgebra:-RandomMatrix( 1600, outputoptions = [datatype = float[8]] ):

time( M.M ); # see text above for timings

Linux and Macintosh OS X

  

Maple 9.5 for Linux and Macintosh OS X uses tuned versions, from ATLAS, of a select group of CLAPACK functions related to hardware floating-point linear algebra computations, such as LUDecomposition and LinearSolve.

  

The following example takes approximately 0.66 seconds in Maple 9 and 0.40 seconds in Maple 9.5  on a 2.4GHz P4 machine running Linux. On a G5 machine running OS X 10.3, this example takes approximately 0.43 seconds in Maple 9 and 0.25 seconds in Maple 9.5.

M := LinearAlgebra:-RandomMatrix( 1000, outputoptions = [datatype = float[8]] ):

V := LinearAlgebra:-RandomVector( 1000, outputoptions = [datatype = float[8]] ):

time( LinearAlgebra:-LinearSolve( M, V ) ); # see text above for timings

See Also

copyright

Index of New Maple 9.5 Features