Computer Algebra Commands

For HP48 Version 3.3 revision 3 oct 20,2001

By: Jaime Fernando Meza meza

http://www.geocities.com/hp4x/cashp48/computeralgebracommandshp48g.htm

http://www.geocities.com/hp4x/cashp48/font8forcashp48.zip

CAShp48.LIB alone works jointly with:

(Library 909 Algebra48 version 4.2),

(Library 913 Symbolic integration version 0.1)

(Library 1494 PolySymbolic version 6.6) and

(Library 905 QPI version 5.3)

CASHP48.LIB Contains a large number of computer algebra commands similar to those in the (HP49 version 1.19-5), (DERIVE version 5.04)(www.derive.com), Texas Instruments(TI89/TI92plus) AND (MATLAB version 6.0)

Algebra:

{ SIMPLIFY CSIMPLIFY FACTOR EXACT APPROXIMATE SUBST SOLVE SOLVEVX TABVAL }

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Arithmetic: { PROPFRAC FACTORS PROPFRAC2}

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Arithmetic integer:

{ ISPRIME? PREVPRIME NEXTPRIME MOEBIUS DIVISORS PRIMEDIVIS PERFEC? }

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Arithmetic Modulo:

{ MODSTO ADDTMOD SUBTMOD MULTMOD DIVMOD POWMOD INVMOD }

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Calculus:

{ INTGRS DERIVS GRADIENT DIV CURL LAPLACIAN POTENTIAL VPOTENTIAL HESS LAGRANGE JACOBIAN DERIVN WRONSKYAN ARCLEN FMATRIX MCOS MSIN MEXP BESSEL }

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Polynomial Function:

{ FXND P�COEF C�POL C�FRAT REORDER P�ROOT R�POL P�MROOT MULT2POL DIV2POL QUOT REMAINDER PROPFRAC PARTFRAC CPARTFRAC FACTOR GCD LCM PCAR RANDPOLY PTCHEBYCHEV PUCHEBYCHEV PLEGENDRE PALEGENDRE PHERMITE PLAGUERRE PGLAGUERRE PSPHERICALHARM POLEVALM HORNER }

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Matrices, vector and Linear Algebra

Create:

{ NEWMAT NEWVECTOR RANDMAT RANDVECTOR RANDCMAT RANDCVECTOR RANDSYSMAT RANDASYSMAT RANDUPPERMAT RANDLOWERMAT RANDSTOCMAT AUGMENT APPENDCOL MDIAG LCXM MAKEMAT HILBERT MVANDERMONDE VECTOR }

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Operations:

{ LIST�MAT MAT�LIST AXL TRAN DIMMAT MAPRG XNUM XEVAL AXM DIST ANGLE DANGLE VPRBA PRYBA HADAMARD .* MDIV ./ .^ MADD .+ MSUB .- MULT POWMAT INVS MINV ADJOINT NORMAT i_ j_ k_ }

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Quadratic form: AXQ QXA

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Vector: { ABS2 DOTP CROSSP PRODH UNITV COSDIR DIST }

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Factr: { lu qr CHOLESKY }

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Linear Systems: { LINSOLVE REF oref CRAMER MINOR COFACT }

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Eigenvector: { PCAR PMINI IMAGE KER EGBS JORDAN }

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Col AND Row: { COLDIM -COL COL-2 ROW-2 ROWSAWP }

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Complex:

{ RE2 IM2 C�R2 R�C2 ABS2 ARG2 SIGN2 NEG2 CONJ2 C�Rxy CEILING }

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Statistics:

{ SORTA SORTD CUMSUM CUMPROD %TILE MEAN2 MEDIAN ONETWOVAR FACTORIAL }

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Differencial equations: { DESOLVE LAPLACE ILAPLACE }

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List: LAND LOR LXOR LDIFFERENCE LSAME LDEL LSWAP LDELMULT

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Misc:

MAKEVX CATALOG CASCMD AUTHOR A�RPN pi FIBONACCI �PRg OBj� �Id BZIP UNZIP PREFIXES TYPE? VXPURGE CRCASDIR TMENU2 PROMPT2 VER up EXIT }

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Objects:

[ Left vector/matrix delimiter

] right vector/matrix delimiter

{ Left vector/matrix delimiter

} right vector/matrix delimiter

{ Left list delimiter

} right list delimiter

' Left and right expression delimiter

"Left and right string delimiter to represent long precision floating point numbers or unlimited precision binary integer (hexstring).

A*X=b

[[ x11 x12 ... x1n ] [[ X1 ] [[ X1 ]

[ x21 x22 ... x2n ] * [ Xi ] = [ Xi ]

[ xm1 xm2 ... xmn ]] [ Xn ]] [ Xn ]]

MATRIZ(A) Vector columns(X,b)

A: Numeric Vector

a vector with elements numeric can be entered by typing delimiters of the form:

where the Xi are the elements numeric of the vector. The elements must be separated by spaces, and the vector must be delimited using square brackets [ Xi ], or using square braces { Xi }.

A1: vector column

[ X1 X2 Xi ... Xn ] or [[ X1 ]

[ Xi ]

[ Xn ]]

Example #1: [ 18 24 4 ] (vector column)

Example #2: [[ 18 ]

[ 24 ]

[ 4 ]] (vector column)

Example #3: { 18 24 4 } (vector column)

Example #4: {{ 18 }

{ 24 }

{ 4 }} (vector column)

A2: vector row

[[ X1 X2 Xi ... Xn ]]

Example #5: [[ 18 24 4 ]] (vector row)

B: Symbolic Vector

A vector with elements symbolic or numeric can be entered by typing delimiters of the form:

where the 'Xi' are the elements symbolic or numeric of the vector. The elements must be separated by spaces, and the vector must be delimited using square braces { 'Xi' }.

A1: vector column

{ 'X1' 'X2' 'Xi' ... 'Xn' } or {{ 'X1' }

{ 'X2' }

{ 'Xn' }}

Example #1: { '2*x' –5 '‡' 1.5 'x^2'}

Example #2: { X Y Z } or {{ x }

{ y }

{ z }}

A2: vector row

{{ 'X1' 'X2' 'Xi' ... 'Xn' }}

Example #3: {{ X Y Z }}

C: Numeric Array

A matrix with elements numeric can be entered by typing delimiters of the form:

[[ x11 x12 ... x1n ]

[ x21 x22 ... x2n ]

...

[ xm1 xm2 ... xmn ]]

where the xij are the elements numeric of the matrix. The elements of a row must be separated by spaces, and the matrix must be delimited square brackets [ xij ], or using square braces { xij }.

Internally matrices are stored as vectors of vectors row.

Example #1: [[ 2 4 6 ]

[ 4 5 6 ]

[ 3 1 -2 ]]

Example #2: {{ 2 4 6 }

{ 4 5 6 }

{ 3 1 -2 }}

D: Symbolic Array

A matrix with elements symbolic or numeric can be entered by typing delimiters of the form:

{{ 'x11' 'x12' ... 'x1n' }

{ 'x21' 'x22' ... 'x2n' }

...

{ 'xm1' 'xm2' ... 'xmn' }}

where the xij are the elements numeric of the matrix. The elements of a row must be separated by spaces, and the matrix must be delimited square brackets [ xij ], or using square braces { xij }.

Internally matrices are stored as vectors of vectors row.

Example #1: {{ a 4 }

{ b e }

{ c 5 }}

E: Algebraic expression: '(X^2+2*X+1)/(X+1)'

F: Variables: 'X' 'S' 't' 'XY' ... etc

G: To represent long precision floating point numbers "130529377836972488251268578591"

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EXPAND or SIMPLIFY are similar to expand of the (TI92+) or SIMPLIFY of the HP49

Description: Expands and simplifies an object or a list of objects.

Access: Library 697: CAS48 SIMPLIFY

Example #1: Simplify the following algebraic expression:

Input:

Level 1: '(X^2+2*X+1)/(X+1)' SIMPLIFY [ENTER]

Output:

Level 1: 'X+1'

input: output:

�

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Example #2: Simplify the following algebraic expression:

Input:

Level 1: '3*X*(X^2-3*X+1)*(X^4+X-5)*(5*X-7)^2' SIMPLIFY [ENTER]

Output:

Level 1: '75*X^9-435*X^8+852*X^7-576*X^6-663*X^5+3027*X^4-4911*X^3+3402*X^2-735*X'

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Example #3: Simplify the following Symbolic Matrix of expressions:

Input:

Level 1: {{ '(X+1)^2' }

{ 'COS(‡)' }} RAD SIMPLIFY [ENTER]

Output:

Level 1: {{ 'X^2+2*X+1' }

{ -1 }}

input: output:

�

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Example #4:

input: output:

�

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'-b*d+a*d*(0,1)+b*i*c+a*c' CSIMPLIFY

'a*c-b*d+(a*d+b*c)*i'

'INV(a+b*i)' CSIMPLIFY

'a/(a^2+b^2)-b/(a^2+b^2)*i'

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FACTOR similar to factor of the (TI92+) or FACTOR of the HP49

Description: Factorizes a polynomial or Symbolic Matrix.

Access: Library 697: CAS48 FACTOR

Example #1: Factorize the following polynomial:

Input:

Level 1: 'X^2+5*X+6' FACTOR

Output:

Level 1: '(X+2)*(X+3)'

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Example #2: Factorize the following expression:

Input:

Level 1: '75*X^9-435*X^8+852*X^7-576*X^6-663*X^5+3027*X^4-4911*X^3+3402*X^2-735*X' FACTOR

Output:

Level 1: '3*X*(X^2-3*X+1)*(X^4+X-5)*(5*X-7)^2'

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Example #3: Factorize the following integer:

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Example #4: Factorize the following long precision numbers:

"130529377836972488251268578591" FACTOR

'2647*3691*5113*11779*398609*556517681' APPROXIMATE

'a^3*X^2+a*X^2+a^3+a' FACTOR

'a*(a^2+1)*(X^2+1)'

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'a^3*X^2-a*X^2-a^3+a' FACTOR

'a*(a-1)*(a+1)*(X-1)*(X+1)'

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'P*T^2-P+1-T^2' FACTOR

'(T-1)*(T+1)*(P-1)'

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EXACT similar to exact of the (TI92+) or XQ of the HP49

Description: Uses Exact mode arithmetic

Access: Library 697: CAS48 EXACT

Example #1: Express in exact format.

Input:

Level 1: .25 EXACT

Output:

Level 1: '1/4'

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Example #2: Express in exact format.

Input:

Level 1: {{ 2.5 2.71828182846 1.5707963268 }

{ (0,1) 1.41421356237 0.33 }} EXACT

Output:

Level 1: {{ '5/2' 'EXP(1)' '1/2*‡' }

{ 'i' 'ƒ2' '33/100' }}

See also: XQ APPROXIMATE XQ� ~~ XNUM

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APPROXIMATE similar to approx of the (TI92+)

Description: Returns the evaluation of expression as a decimal value

where each element has been evaluated to a decimal value.

Access: Library 697: CAS48 APPROXIMATE

Example #1:

Input:

Level 1: '1/4'

Output:

Level 1: .25

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Example #2:

Input:

Level 1: {{ '5/2' 'EXP(1)' '1/2*‡' }

{ 'i' 'ƒ2' '33/100' }} APPROXIMATE

Output:

Level 1: {{ 2.5 2.71828182846 1.5707963268 }

{ (0,1) 1.41421356237 0.33 }}

See also: XQ� ~~ XQ

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AXL similar to list�mat or mat�list of the (TI92+)or AXL of the HP49

Description: Converts a Symbolic/numeric Matrix {{ xi }} to a numeric Matrix [[ xij ]].

Access: Library 697: CAS48 AXL

Example #1: Convert the following numeric Matrix [[ xij ]] to a numeric Matrix {{ xij }}

Input:

Level 1: [[ 2 4 6 ]

[ 4 5 6 ]

[ 3 1 -2 ]] AXL

Output:

Level 1: {{ 2 4 6 }

{ 4 5 6 }

{ 3 1 -2 }}

Example #2: Convert the following numeric Vector [ xi ] to a numeric Vector { xi }

Input:

Level 1: [ 18 24 4 ] AXL

Output:

Level 1: { 18 24 4 }

Example #3: Convert the following numeric Matrix {{ xi }}to a numeric Matrix [[ xij ]]

Input:

Level 1: {{ 2 4 6 }

{ 4 '10/2' 6 }

{ 3 1 -2 }} AXL

Output:

Level 1: [[ 2 4 6 ]

[ 4 5 6 ]

[ 3 1 -2 ]]

Example #4:

Convert the following numeric Vector { xi } to a Vector[ xi ]

Input:

Level 1: { '5/2' '7/2' '‡' } AXL

Output:

Level 1: [ 2.5 3.5 3.14159265359 ]

See also: LISTMAT MATLIST

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TRAN similar to T of the (TI92+) or TRN of the(HP48 but TRAN accepts symbolic arguments) or TRAN of the HP49

Description: Returns the transpose of matrix1.

Access: Library 697: CAS48 TRAN

Example #1: Transpose the following Matrix

Input:

Level 1: {{ a b }

{ c d }} TRAN

Output:

Level 1: {{ a c }

{ b d }}

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Example #2: Transpose the following vector row.

Input:

Level 1: {{ 18 24 4 }} TRAN

Output:

Level 1: {{ 18 }

{ 24 }

{ 4 }}

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Example #3: Transpose the following vector column.

Input:

Level 1: { 18 24 4 } TRAN

Output:

Level 1: {{ 18 24 4 }}

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Example #4: Transpose the following vector column.

Input:

Level 1: {{ A }

{ B }

{ C }} TRAN

Output:

Level 1: {{ A B C }}

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Example #5: Transpose the following vector column.

Input:

Level 1: [ 18 24 4 ] TRAN

Output:

Level 1: {{ 18 24 4 }}

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ROWSWAP similar to rowSwap of the (TI92+) or RSWP of the(HP48 but RSWP2 accepts symbolic arguments)

Description: Returns the ROW SWAP. Permutation of a Matrix, the rowi and rowj rows are interchanged

Access: Library 697: CAS48 RSWP2

Example #1:

Input:

Level 3: {{ a d }

{ b e }

{ c f }}

Level 2: 2

Level 1: 3 ROWSWAP

Output:

Level 1: {{ a d }

{ c f }

{ b e }}

See also: RSWP

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C�Rxy similar to P>Rx of the (TI92+) or P>Ry of the (TI92+) or C�R of the HP48 (but C�Rxy accepts symbolic arguments)

Returns the equivalent x coordinate and y coordinate of a complex number.

Example #1: in polar form

angle mode: 1: « DEG CYLIN » [ENTER]

Input:

Level 1: (4,€60) C�Rxy

Output:

Level 1: { 2 '2*ƒ3' }

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Example #2: in rectangular form

angle mode 1: « DEG RECT » [ENTER]

Input:

Level 1: (3,4) C�Rxy

Output:

Level 1: { 3 4 }

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Example #3: in symbolic form

Input:

Level 1: '3+4*i' C�Rxy

Output:

Level 1: { 3 4 }

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Example #4: in vector form

angle mode 1: « DEG RECT » [ENTER]

Input:

Level 1: [ 3 4 ] C�Rxy

Output:

Level 1: { 3 4 }

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Example #5: number real

Input:

Level 1: 3 C�Rxy

Output:

Level 1: { 3 0 }

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Example #6: number complex

Input:

Level 1: '4*i' C�Rxy

Output:

Level 1: { 0 4 }

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Example #7: in form of symbolic expression

Input:

Level 1: 'i^2+2*i+i+3+i' C�Rxy

Output:

Level 1: { 2 4 }

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Example #8: lists of complex expressions

To calculate this command in each element you should apply successively << C�Rxy >> with the command MAPRG.

{(-3,€ ‡/3) (10,€ ‡/4) (3,€0)}

angle mode 1: « RAD CYLIN » [ENTER]

Input:

Level 1: { (3,€-2.09439510239) (10,€-.785398163398) 1.3 }

« C�Rxy » MAPRG

Output:

Level 1: {{ '-3/2' '-(3/2*ƒ3)'}

{ '5*ƒ2' '-(5*ƒ2)' }

{ '13/10' 0 }}

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Example #9: in form of complex matrix

To calculate this command in each element you should apply successively << C�Rxy >> with the command MAPRG.

Input:

Level 1: {{ 'a+i*b' 3 }

{ c 'i' }} « C�Rxy » MAPRG

Output:

Level 1: { {{ a b }

{ 3 0 }}

{{ c 0 }

{ 0 1 }}}

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Example #10:

'(a+b*i)/(c+d*i)' C�Rxy

{ '(a*c+b*d)/(c^2+d^2)' '(-(a*d)+b*c)/(c^2+d^2)' }

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Example #11:

'(a+b*i)*(c+d*i)' C�Rxy

{ 'a*c-b*d' 'a*d+b*c' }

See also: C�R2

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C�R2 similar to C�R of the (HP48) or C�Rxy but C�R2 accepts symbolic arguments

Example #1: in polar form

angle mode: « DEG CYLIN »

Input:

Level 1: (4,€60) C�R2

Output:

Level 2: 2

Level 1: '2*ƒ3'

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Example #2: in symbolic form

Input:

Level 1: '3+4*i' C�R2

Output:

Level 2: 3

Level 1: 4

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Example #3: in form of symbolic expression

Input:

Level 1: 'i^2+2*i+i+3+i' C�R2

Output:

Level 2: 2

Level 1: 4

See also: C�R

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R�C2

Description: similar to R�C of the (HP48) but R�C2 accepts symbolic arguments

process inverse to the previous one

See also: R�C

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RE2

Description: similar to P>Rx of the (TI92+) or real of the (TI92+) or RE of the (HP48 but it accepts symbolic arguments)

returns the real part of the argument or returns the equivalent x coordinate of a complex number.

Access: Library 696: LINEAL DOTS CATALOG

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(rExpression € •Expression) � expression

Example #1: in polar form

angle mode: « DEG CYLIN »

Input:

Level 1: (4,€60) RE2

Output:

Level 1: 2

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expression � expression

Example #2: in symbolic form

Input:

Level 1: 'z' RE2

Output:

Level 1: z

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expression � expression

Example #3: in symbolic form

Input:

Level 1: 'x+y*i' RE2

Output:

Level 1: x

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list � list

Example #4: lists of complex expressions

To calculate this command in each element you should apply successively << RE2 >> with the command MAPRG.

Input:

Level 1: { 'a+b*i' 3 i } « RE2 » MAPRG

Output:

Level 1: { a 3 0 }

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matrix � matrix

Example #5: in form of complex matrix

To calculate this command in each element you should apply successively << RE2 >> with the command MAPRG.

Input:

Level 1: {{ 'a+i*b' 3 }

{ c 'i' }} « RE2 » MAPRG

Output:

Level 1: {{ a 3 }

{ c 0 }}

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list � list

Example #6: lists of complex expressions

To calculate this command in each element you should apply successively << RE2 >> with the command MAPRG.

{(-3,€ ‡/3) (10,€ ‡/4) (3,€0)}

angle mode: « RAD CYLIN »

Input:

Level 1: { (3,€-2.09439510239) (10,€-.785398163398) 1.3 } « RE2 » MAPRG

Output:

Level 1: { '-3/2' '5*ƒ2' '13/10' }

See also: RE

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IM2 similar to P>Ry of the (TI92+) or imag of the (TI92+) or IM of the (HP48 but it accepts symbolic arguments)

returns the imaginary part of the argument or returns the equivalent y coordinate of a complex number.

Example #1: in polar form

angle mode: « DEG CYLIN »

Input:

Level 1: (4,€60) IM2

Output:

Level 1: '2*ƒ3'

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Example #2: in symbolic form

Input:

Level 1: '3+4*i' IM2

Output:

Level 1: 4

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Example #3: lists of complex expressions

To calculate this command in each element you should apply successively << IM2 >> with the command MAPRG.

Input:

Level 1: { 'a+b*i' 3 i } « IM2 » MAPRG

Output:

Level 1: { b 0 1 }

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Example #4: in form of complex matrix

To calculate this command in each element you should apply successively << IM2 >> with the command MAPRG.

Input:

Level 1: {{ 'a+i*b' 3 }

{ c 'i' }} « IM2 » MAPRG

Output:

Level 1: {{ b 0 }

{ 0 1 }}

See also: IM

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MULT

Description: similar to * of the (TI92+)

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matrix1( m1 x n1 ) MULT matrix2( m2 x n2 ) � matrix

Returns the matrix product of matrix1 and

matrix2.

The number of columns in matrix1 must equal

the number of rows in matrix2. (n1 = m2)

Note: e=2.71828182846

Example #1:

Input:

Level2: {{ 1 2 3 }

{ 4 5 6 }}

Level1: {{ a d }

{ b E }

{ c f }}

Output:

Level1: {{ 'a+2*b+3*c' 'd+3*f+2*E' }

{ '4*a+5*b+6*c' '4*d+6*f+5*E' }}

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expression MULT list1 � list

list1 MULT expression � list

Returns a list containing the products of

expression and each element in list1.

Example #3:

Input:

Level2: '‡'

Level1: { 4 5 6 } MULT

Output:

Level1: { '4*‡' '5*‡' '6*‡' }

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expression MULT matrix1 � matrix

matrix MULT expression � matrix

Returns a matrix containing the products of

expression and each element in matrix1.

Example #4:

Input:

Level2: {{ 1 0 0 }

{ 0 1 0 }

{ 0 0 1 }}

Level1: x MULT

Output:

Level1: {{ x 0 0 }

{ 0 x 0 }

{ 0 0 x }}

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list MULT list � expression

Returns the dot product of two lists.

Both must be column vectors.

Example #5: See also: DOTP

Input:

Level2: { a b c }

Level1: { d E f } MULT

Output:

Level1: 'E*b+a*d+c*f'

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Example #6: See also: DOTP

Input:

Level2: { 2 a 5 }

Level1: { '2*a' 3 -1 } MULT

Output:

Level1: '7*a-5'

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DOTP

Description: similar to DOTP of the (TI92+) or DOT of the (HP48 but DOTP accepts symbolic arguments).

The dot product of two vectors (also called the scalar product) is the sum of the products of corresponding elements of the vectors. The dimensions of the vectors must be the same. Note that the result is a scalar.

Access: Library 696: LINEAL DOTP CATALOG

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list DOTP list � expression

Returns the dot product of two lists.

Both must be column vectors.

Example #1:

Input:

Level2: { 1 2 }

Level1: { 5 6 } DOTP

Output:

Level1: 17

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Example #2

Input:

Level2: {{ a b }

{ c d }}

Level1: { 2 3 } DOTP

Output:

Level1: { '2*a+3*b' '2*c+3*d' }

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Example #3: See also: PRODH

Input:

Level2: { (1,0) (2,-1) }

Level1: { (4,-1) (0,2) } DOTP

Output:

Level1: '6+3*i'

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Example #4:

Input:

Level2: { (1,0) (2,-1) }

Level1: { (4,-1) (0,2) } PRODH

Output:

Level1: '2+3*i'

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CROSS2

Description: Returns the cross product of two vectors C = A × B

Access: Library 696: LINEAL CROSS2

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list1 CROSS2 list2 � list

Returns the cross product of list1 and list2

list1 and list2 must have equal dimension, and

the dimension must be either 3.

Example #1:

Input:

Level2: { 1 3 4 }

Level1: { 2 7 -5 } CROSS2

Output:

Level1: [ -43 13 1 ]

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Example #2:

Input:

Level2: { 1 3 4 }

Level1: { a b c } CROSS2

Output:

Level1: { '-(3*b)+2*c' '3*a-c' '-(2*a)+b' }

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Example #3

Input:

Level2: { 1 2 0 }

Level1: { a b 0 } CROSS2

Output:

Level1: { 0 0 '-(2*a)+b' }

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vector1 CROSS2 vector2 � vector

Returns the cross product of vector1 and vector2

list1 and list2 must have equal dimension, and

the dimension must be either 3.

Example #4

Input:

Level2: [ 1 2 3 ]

Level1: [ 4 5 6 ] CROSS2

Output:

Level1: [ -3 6 -3 ]

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ABS2

Description: Similar to abs of the (TI92+) or ABS of the (HP48 but ABS2 accepts symbolic arguments).

Returns the absolute value of the argument.

If the argument is a complex number, returns

the magnitude (modulos).

Access: Library 697: CAS48/ABS2 or CATALOG

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expression � expression

Example #1: in polar form

angle mode: « DEG CYLIN »

Input:

Level 1: (5,€53.1301023542) DEG CYLIN ABS2

Output:

Level 1: 5

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Example #2: in rectangular form

angle mode: « DEG RECT »

Input:

Level 1: (3,4) ABS2

Output:

Level 1: 5

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Example #3: in symbolic form

Input:

Level 1: '3+4*i' ABS2

Output:

Level 1: 5

___________________________________

Example #4: in vector form

angle mode: « DEG RECT »

Input:

Level 1: [ 3 4 ] ABS2

Output:

Level 1: 5

___________________________________

Example #5: number real

Input:

Level 1: -3 ABS2

Output:

Level 1: 3

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Example #6: in form of symbolic expression

Input:

Level 1: 'i^2+2*i+i+3+i' ABS2

Output:

Level 1: '2*ƒ5'

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Example #7:

Input:

Level 1: '3*x+4*i' ABS2

Output:

Level 1: 'ƒ(9*x^2+16)'

___________________________________

list � list

Example #8:

Input:

Level 1: { 3 4 } ABS2

Output:

Level 1: 5

___________________________________

Example #9:

Input:

Level 1: { 'COS(t)' 'SIN(t)' } ABS2

Output:

Level 1: 1

___________________________________

Example #10:

Input:

Level 1: { 3 4 'ƒ11' 'ƒ13' } ABS2

Output:

Level 1: 7

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SIGN2

Access: Library 697: CAS48 SING2 / CATALOG

Description: Returns the sing of the argument.

If the argument is a real expression:

Returns 1 if expression1 is positive.

Returns -1 if expression1 is negative.

SING2(0) returns 0

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expression � vector

For a expression returns A row -unit vector,

expression1/(abs(expression1)).

depending on the form of the expression.

Example #1:

Input:

Level 1: (3,4) SIGN2

Output:

Level 1: { '3/5' '4/5' }

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Example #2:

Input:

Level 1: { 3 4 } SIGN2

Output:

Level 1: { '3/5' '4/5' }

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Example #3:

Input:

Level 1: [ 3 4 ] SIGN2

Output:

Level 1: { '3/5' '4/5' }

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Example #4:

Input:

Level 1: 'X+Y*i' SIGN2

Output:

Level 1: { 'X/ƒ(X^2+Y^2)' 'Y/ƒ(X^2+Y^2)' }

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Example #5:

Input:

Level 1: '3*z+4*i' SIGN2

Output:

Level 1: { '3*z/ƒ(9*z^2+16)' 'INV(1/4*ƒ(9*z^2+16))' }

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Example #6:

Input:

Level 1: { 'COS(t)' 'SIN(t)' } SIGN2

Output:

Level 1: { 'COS(t)' 'SIN(t)' }

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Example #7:

Input:

Level 1: { 3 4 'ƒ11' 'ƒ13' } SIGN2

Output:

Level 1: { '3/7' '4/7' '1/7*ƒ11' '1/7*ƒ13' }

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Example #8:

Input:

Level 1: 'i^2+2*i+i+3+i' SIGN2

Output:

Level 1: { '1/5*ƒ5' '2/5*ƒ5' }

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Example #9:

Input:

Level 1: { a b c } SIGN2

Output:

Level 1: { 'a/ƒ(a^2+b^2+c^2)' 'b/ƒ(a^2+b^2+c^2)' 'c/ƒ(a^2+b^2+c^2)' }

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CONJ2

Description: Returns the complex conjugate of the

argument.

To calculate this command in each element you should apply successively « CONJ2 » with the command MAPRG.

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Example #1:

Input:

Level 1: Z CONJ2

Output:

Level 1: Z

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Example #2:

Input:

Level 1: 'X-Y*i' CONJ2

Output:

Level 1: 'X+Y*i'

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Example #3:

Input:

Level 1: {{ 2 '1-3*i' }

{ '-i' -7 }} « CONJ2 » MAPRG

Output:

Level 1: {{ 2 '1+3*i' }

{ 'i' -7 }}

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'2*i' RAD ANGLE

'‡/2'

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'1+i' RAD ANGLE

'1/4*‡'

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'x+y*i' RAD ANGLE

'1/2*‡*SIGN(y)-ATAN(x/y)'

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{{ a b }

{ c d }} NORMAT

'ƒ(a^2+b^2+c^2+d^2)'

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{{ 1 2 }

{ 3 4 }} NORMAT

'ƒ30'

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{ 'EXP(3*t)' 'EXP(-3*t)' t }

t WRONSKYAN

{{ 'EXP(3*t)' 'EXP(-3*t)' t }

{ '3*EXP(3*t)' 'INV(-(1/3*EXP(3*t)))' 1 }

{ '9*EXP(3*t)' 'INV(1/9*EXP(3*t))' 0 }}

:DET=: '54*t'

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{ 'EXP(t)' 'EXP(2*t)' 'EXP(3*t)' }

t WRONSKYAN

{{ 'EXP(t)' 'EXP(2*t)' 'EXP(3*t)' }

{ 'EXP(t)' '2*EXP(2*t)' '3*EXP(3*t)' }

{ 'EXP(t)' '4*EXP(2*t)' '9*EXP(3*t)' }}

:DET=: '2*EXP(6*t)'

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.+ similar to .+ of the (TI92+)

{ 1.9 'EXP(1)' 'COS(t)^2' }

{ 0.1 'i' 'SIN(t)^2' } .+

{ 3 'i+EXP(1)' 1 }

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{{ a b }

{ c E }}

[[ 1 0 ]

[ 0 1 ]] MADD or .+

{{ 'a+1' b }

{ c 'E+1' }}

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{{ a 2 }

{ b 3 }}

{{ c 4 }

{ d 5 }} MSUB or .-

{{ 'a-c' -2 }

{ 'b-d' -2 }}

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{ 'EXP(X)' 'i' '‡' }

{ 'INV(EXP(X))' 'i' .5 } .*

{ 1 -1 '1/2*‡' }

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{{ a 2 }

{ b 3 }}

{{ c 4 }

{ d 5 }} HADAMARD or .*

{{ 'a*c' 8 }

{ 'b*d' 15 }}

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{ 'ƒX' X }

{ X 'ƒX' } ./

{ 'INV(ƒX)' 'ƒX' }

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{{ a 2 }

{ b 3 }}

{{ c 4 }

{ d 5 }} MDIV or ./

{{ 'a/c' '1/2' }

{ 'b/d' '3/5' }}

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{ 'x+y' 'e' 'EXP(y)' }

{ 2 'LN(z)' 'LN(x)' } .^

{ 'x^2+2*x*y+y^2' z 'x^y' }

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{{ a 2 }

{ b 3 }}

{{ c 4 }

{ 5 d }} .^

{{ 'a^c' 16 }

{ 'b^5' '3^d' }}

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AUGMENT similar to AUGMENT of the (TI92+)

{{ 1 2 3 }}

{ X } AUGMENT

{{ 1 2 3 X }}

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{{ 1 2 }

{ 3 X }}

{ 5 6 } AUGMENT

{{ 1 2 5 }

{ 3 X 6 }}

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{{ 1 2 }

{ 3 X }}

{{ 5 }

{ 6 }} AUGMENT

{{ 1 2 5 }

{ 3 X 6 }}

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{{ 1 2 }

{ 3 X }}

{{ 1 0 }

{ 0 1 }} APPENDCOL

{{ 1 2 }

{ 3 X }

{ 1 0 }

{ 0 1 }}

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2 RANDMAT

[[ 3 -4 ]

[ 6 -1 ]]

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3 RANDVECTOR

[ 3 -8 -5 ]

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RANDPOLY similar to RANDPOLY of the (TI92+)

5 RANDPOLY

'7*X^5-X^4+7*X^3-5*X^2+2*X+2'

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{{ 1 2 }

{ a b }} 2 POWMATRIX

{{ '2*a+1' '2*b+2' }

{ 'a*b+a' '2*a+b^2' }}

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{{ 1 2 }

{ a b }} -1 POWMATRIX

{{ '-b/(2*a-b)' 'INV(a-1/2*b)' }

{ 'a/(2*a-b)' 'INV(-(2*a)+b)' }}

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{{ 1 2 }

{ a b }} INVS

{{ '-b/(2*a-b)' 'INV(a-1/2*b)' }

{ 'a/(2*a-b)' 'INV(-(2*a)+b)' }}

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[[ 1 2 ]

[ 3 4 ]] -2 POWMATRIX

{{ '11/2' '-(5/2)' }

{ '-(15/4)' '7/4' }}

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CUMSUM similar to CUMSUM() of the (TI92+)

{ 1 2 3 4 } CUMSUM

{ 1 3 6 10 }

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{ a b c } CUMSUM

{ a 'a+b' 'a+(b+c)' }

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{ X Y Z } MDIAG

{{ X 0 0 }

{ 0 Y 0 }

{ 0 0 Z }}

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VECTOR similar to VECTOR of the (DERIVE V5.02)

VECTOR(u, k, n) simplifies to a vector of n elements

generated by simplifying the expression u(k)

with the variable k stepping from begin through n

in steps of size m.

example #1:

VECTOR(x^2,x,1,5,1)

where

level 5: x^2 expression

level 4: x variable

level 3: 1 begin

level 2: 5 end(n)

level 1: 1 step(m)

'x^2' x 1 5 1 VECTOR

{{ 1 }

{ 4 }

{ 9 }

{ 16 }

{ 25 }}

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example #2:

VECTOR(j!,j,0,4,1)

'j!' j 0 4 1 VECTOR

[[ 1 ]

[ 1 ]

[ 2 ]

[ 6 ]

[ 24 ]]

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VECTOR(SIN(z),z,0, ‡/4,0.2)

'SIN(z)' z 0 '‡/4' .2 VECTOR

[[ 0 ]

[ .198669330795 ]

[ .389418342309 ]

[ .564642473395 ]]

{{ 0 }

{ 'SIN(1/5)' }

{ 'SIN(2/5)' }

{ '3/5' } }

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example #4:

VECTOR({x,x^2,x^3},x,1,4,1)

{ x 'x^2' 'x^3' } x 1 4 1 VECTOR

[[ 1 1 1 ]

[ 2 4 8 ]

[ 3 9 27 ]

[ 4 16 64 ]]

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example #5:

VECTOR({x+y,x^2},x,1,4,1.5)

{ 'x+y' 'x^2' } x 1 4 1.5 VECTOR

{{ '1+y' 1 }

{ '2.5+y' 6.25 }

{ '4+y' 16 }}

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'GAMMA(X)'

X

1

10

1 VECTOR

[[ 1 ]

[ 1 ]

[ 2 ]

[ 6 ]

[ 24 ]

[ 120 ]

[ 720 ]

[ 5040 ]

[ 40320 ]

[ 362880 ]]

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{ X «X FIBONACCI» }

X

1

9

1 VECTOR

[[ 1 1 ]

[ 2 1 ]

[ 3 2 ]

[ 4 3 ]

[ 5 5 ]

[ 6 8 ]

[ 7 13 ]

[ 8 21 ]

[ 9 34 ]]

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{ X «X FACTOR» }

X

1

10

1 VECTOR

{{ 1 1 }

{ 2 2 }

{ 3 3 }

{ 4 '2^2' }

{ 5 5 }

{ 6 '2*3' }

{ 7 7 }

{ 8 '2^3' }

{ 9 '3^2' }

{ 10 '2*5' }}

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SORTA similar to SortA() of the (TI92+)

Sorts the elements of the first argument in

ascending order.

[ 2 1 4 3 ] SORTA

{ 1 2 3 4 }

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{ e '‡' 'ƒ2' 3 } SORTA

{ 'ƒ2' 'EXP(1)' 3 '‡' }

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[[ 2 3 ]

[ 3 1 ]

[ 2 1 ]

[ 1 5 ]] SORTA

{{ 1 5 }

{ 2 3 }

{ 2 1 }

{ 3 1 }}

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SORTD similar to SortD() of the (TI92+) Identical to SORTA, except SORTD sorts the elements in descending order.

[ 2 1 4 3 ] SORTD

{ 4 3 2 1 }

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{ A a E C F } SORTD

{ a F E C A }

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'COS(x)'

'x'

0

'‡' ARCLEN

3.82019778903

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[[ 1 5 3 ]

[ 4 2 1 ]

[ 6 -2 1 ]] MCOS

[[ .212493123035 .205063658269 .12138913931 ]

[ .160870605001 .259041970092 3.71256719027E-2 ]

[ .248078814643 -9.01529518783E-2 .218971555953 ]]

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[[ 1 5 3 ]

[ 4 2 1 ]

[ 6 -2 1 ]] MEXP

[[ 782.208648399 559.616918461 456.508698092 ]

[ 680.546446868 488.795481398 396.521387369 ]

[ 524.929209349 371.221785757 307.878642269 ]]

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[[ .2 0 ]

[ 1 -.3 ]

[ .4 -.5 ]] MEDIAN

[ .4 -.3 ]

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[[ 18 12 ]

[ 4 7 ]

[ 3 2 ]

[ 11 1 ]

[ 31 48 ]

[ 20 17 ]] MEDIAN

[ 14.5 9.5 ]

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[ 8 3 1 5 2 ]

50 %TILE

3

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[[ 0 0 ]

[ 1 2 ]

[ 2 3 ]

[ 3 4 ]

[ 4 3 ]

[ 5 4 ]

[ 6 6 ]] ONETWOVAR

{{ :STAT: { X Y }}

{ :…: { 21 22 }}

{ :nStat:{ 7 7 } }

{ :�: { 3 3.14285714286 }}

{ :…²: { 91 90 }}

{ :…xy: { 88 88 }}

{ :min: { 0 0 }}

{ :max: { 6 6 }}

{ :SDEV: { 2.16024689947 1.86445447147 }}

{ :VAR: { 4.66666666667 3.47619047619 }}

{ :PSDEV:{ 2 1.7261494248 }}

{ :PVAR: { 4 2.97959183673 }}}

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PTCHEBYCHEV similar to CHEBYCHEV_T(n, x) of the (DERIVE V5.02)

0

X PTCHEBYCHEV

1

___________________________________

1

X PTCHEBYCHEV

X

___________________________________

6

X PTCHEBYCHEV

'32*X^6-48*X^4+18*X^2-1'

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PUCHEBYCHEV similar to CHEBYCHEV_U(n, x) of the (DERIVE V5.02)

0

X PUCHEBYCHEV

1

___________________________________

1

X PUCHEBYCHEV

'2*X'

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6

X PUCHEBYCHEV

'64*X^6-80*X^4+24*X^2-1'

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5

'COS(X)' PUCHEBYCHEV

'32*COS(X)^5-32*COS(X)^3+6*COS(X)'

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PLEGENDRE similar to LEGENDRE_P(n, x) of the (DERIVE V5.02)

0

X PLEGENDRE

1

___________________________________

1

X PLEGENDRE

X

___________________________________

6

X PLEGENDRE

'1/16*(231*X^6-315*X^4+105*X^2-5)'

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PALEGENDRE similar to ASSOCIATED_LEGENDRE_P(n, m, x) of (DERIVE V5.02)

4

2

X PALEGENDRE

'15/2*(7*X^2-1)*(1-SQ(X))'

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0

X PHERMITE

1

___________________________________

1

X PHERMITE

'2*X'

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6

X PHERMITE

'64*X^6-480*X^4+720*X^2-120'

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PLAGUERRE similar to LAGUERRE_L(n, x) of the (DERIVE V5.02)

0

X PLAGUERRE

1

___________________________________

1

X PLAGUERRE

'-X+1'

___________________________________

4

X PLAGUERRE

'1/24*(X^4-16*X^3+72*X^2-96*X+24)' SIMPLIFY

'1/24*X^4-2/3*X^3+3*X^2-4*X+1'

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PGLAGUERRE similar to GENERALIZED_LAGUERRE(n, a, x) of (DERIVE V5.02)

5

2

X PGLAGUERRE

'-(1/120)*(X^5-35*X^4+420*X^3-2100*X^2+4200*X-2520)'

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2

1

'COS(X)' PALEGENDRE

'3*COS(X)*SIN(X)'

___________________________________

2

1

•

Ø PSPHERICALHARM

'-(ƒ(5/(24*‡))*(3*COS(•)*SIN(•)*EXP(i*Ø)))'

_____________________________________________________________

100 MODSTO 2 500 POWMOD

76

___________________________________

2.3 ERF .998856823403

___________________________________

2.3 ERFC 1.14317659736E-3

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PARTFRAC

Description: Performs partial fraction decomposition on a partial fraction.

Access: Library 697: CAS48 PARTFRAC

Input: Level 2: An algebraic expression.

Level 1: A variable(s)

Output: The partial fraction decomposition of the expression.

Example #1: Perform a partial fraction decomposition of the following expression:

Input:

Level 2: '1/(X^2-1)'

Level 1: 'X' PARTFRAC

Output:

Level 1: '1/(2*(X-1))-1/(2*(X+1))'

Input: Output:

Example #2: Perform a partial fraction decomposition of the following expression:

Input:

Level 2: '(10*S+20)/(S^3+4*S^2+5*S)'

Level 1: 'S' PARTFRAC

Output:

Level 1:'4/S-(4*S+6)/(S^2+4*S+5)''4/S+(-2-i)/(S-i+2)+(-2+i)/(S+i+2)'

Input: Output: real complex

'(10*S+20)/(S^3+4*S^2+5*S)'

'S' CPARTFRAC

Output:

Level 1: '4/S+(-2-i)/(S+2-i)+(-2+i)/(S+2+i)'

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SUBST

Description: Substitutes value(s) for a variable in an expression. The value can be numeric or an expression.

Access: Library 697: CAS48 SUBST

Input: Level 2: An expression(s).

Level 1: The value or expression to be substituted.

Output: The expression with the substitution made.

Example #1: Substitute x=z+1 for x in the following expression, and apply the EXPAND command to simplify the result:

Input:

Level 2: 'X^2+3*X+7'

Level 1: 'X=Z+1' SUBST

Output 1:

Level 1: '(Z+1)^2+3*(Z+1)+7' SIMPLIFY

Output 2:

Level 1: 'Z^2+5*Z+11'

Example #2:

Input:

Level 2: { 'X-2*COS(t)' 'Y-2*SIN(t)' 'Z=t' }

Level 1: 't=0' SUBST

Level 1: { 'X-2*COS(0)' 'Y-2*SIN(0)' 'Z=0' } SIMPLIFY

Level 1: { 'X-2' 'Y-0' 'Z=0' }

Example #3:

Input:

Level 2: {{ 'X-2*COS(t)' }

{ 'Y-2*SIN(t)' }

{ 'Z=t' }

{ 'R^2=a^2+b^2' }}

Level 1: { 't=0' 'a=3' 'b=4' } SUBST

Output:

Level 1: {{ 'X-2*COS(0)' }

{ 'Y-2*SIN(0)' }

{ 'Z=0' }

{'R=3^2+4^2' }} SIMPLIFY

{{ 'X-2' }

{ 'Y-0' }

{ 'Z=0' }

{ 'R^2=25' }}

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P�ROOT

Description: For a rational polynomial, returns an Vector(list) of P�ROOT

Access: Library 697: CAS48 P�ROOT

Input: Level 2: A Vector { xi }, Matrix {{ xi }}or rationalpolynomial.

Level 1: A variable

Output: A Vector { xi } of the form {Root1 Root2 . . .}

{ a b c }

X P�ROOT

{ :X1: '-(b/(a*2))+ ƒ(SQ(-(b/(a*2)))-c/a)'

:X2: '-(b/(a*2))- ƒ(SQ(-(b/(a*2)))-c/a)' }

___________________________________

{ 1 (-12,-4) (73,36) (-232,-184) (363,376) (-268,-324) (75,100) }

X P�ROOT

{ 1 1 1 '3-4*i' '3+4*i' '3+4*i' }

See also: PROOT

______________________________________________________________

{ 1 1 1 '3-4*i' '3+4*i' '3+4*i' }

X R�POL

'X^6+(-12,-4)*X^5+(73,36)*X^4+(-232,-184)*X^3+(363,376)*X^2+(-268,-324)*X+(75,100)'

___________________________________

{ 1 (-12,-4) (73,36) (-232,-184) (363,376) (-268,-324) (75,100) }

X P�MROOT

{{ 1 3 }

{ '3-4*i' 1 }

{ '3+4*i' 2 }}

___________________________________

{ 2 '1/3' 'A+B' }

X R�POL

'X^3+(-7/3-(A+B))*X^2+(2/3-7/3*-(A+B))*X+2/3*-(A+B)'

___________________________________

'X^3+(-7/3-(A+B))*X^2+(2/3-7/3*-(A+B))*X+2/3*-(A+B)'

SOLVEVX

{{ 'X-2=0' }

{ '3*X-1=0' }

{ 'X-(A+B)=0' }}

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LINSOLVE

Description: Solves a system of linear equations A*X=b.

Access: Library 697: CAS48 LINSOLVE

Example #1: Solve:

Input(Levels):

3: [[ 2 4 6 ]

[ 4 5 6 ]

[ 3 1 -2 ]]

2: [ 18 24 4 ]

1: { X Y Z } LINSOLVE [ENTER]

Output(Levels):

1: { :X: 4 :Y: -2 :Z: 3 }

___________________________________

[[ 2 4 6 ]

[ 4 5 6 ]

[ 3 1 -2 ]] (3*3)

[[ 18 ]

[ 24 ]

[ 4 ]] (3*1)

{{ X }

{ Y }

{ Z }} (3*1) LINSOLVE [ENTER]

A(3*3)*x(3*1)=b(3*1)

(3*1)=(3*1)

{ :X: 4 :Y: -2 :Z: 3 }

___________________________________

[[ 1 2 ]

[ 3 4 ]]

[[ 1 2 ]

[ -1 -3 ]]

{ X Y } LINSOLVE

{ :X: { -3 -7 }

:Y: { 2 4.5 } }

See also: AX=B

Input: Output:

___________________________________

Example #2:

Solve: ax + by = 1

cx + dy = 2

Input(Levels):

3: {{ a b }

{ c d }}

2: {{ 1 }

{ 2 }}

1: { x y } LINSOLVE [ENTER]

Output(Levels):

1:

{ :x: '(-(2*b)+d)/(a*d-b*c)'

:y: '(2*a-c)/(a*d-b*c)' }

___________________________________

{{ a b }

{ c d }}

{{ 1 u }

{ 2 3 }}

{ x y } LINSOLVE [ENTER]

{ :x: { '(-(2*b)+d)/(a*d-b*c)' '(-(3*b)+d*u)/(a*d-b*c)' }

:y: { '(2*a-c)/(a*d-b*c)' '(3*a-c*u)/(a*d-b*c)' }

_____________________________________________________________

DETS

Description: Returns the determinant of a square matrix with or without symbolic elements.

Access: Library 697: CAS48 DETS

Example #1:

Input(Levels):

1: {{ 'X-5' -8 -16 }

{ -4 'X-1' -8 }

{ 4 4 'X+11' }} DETS

Output(Levels):

1: 'X^3+5*X^2+3*X-9'

Input: Output:

_____________________________________________________________

SOLVE

Description: Solution of systems of nonlinear polynomial equations.

Access: Library 697: CAS48 SOLVE

Input:

Level 2: A Vector { xi } of equations.

Level 1: A Vector { xi } of variables.

Example #1:

{ 'X+Y=3' 'X*Y=2' }

{ X Y } SOLVE

{ 'X-1=0' 'Y-2=0' }

{ 'X-2=0' 'Y-1=0' }

___________________________________

Example #2:

{ 'X+Y+Z=6' 'X*Y+X*Z+Y*Z=11' 'X*Y*Z=6' }

{ X Y Z } SOLVE

{ 'X-1=0' 'Y-2=0' 'Z-3=0' }

{ 'X-2=0' 'Y-1=0' 'Z-3=0' }

{ 'X-1=0' 'Y-3=0' 'Z-2=0' }

{ 'X-3=0' 'Y-1=0' 'Z-2=0' }

{ 'X-2=0' 'Y-3=0' 'Z-1=0' }

{ 'X-3=0' 'Y-2=0' 'Z-1=0' }

___________________________________

Example #3:

{ 'X+Y=-b' 'X*Y=c' }

{ X Y } SOLVE

{ 'X+Y+b=0' 'Y^2+b*Y+c=0' }

___________________________________

Example #4: Solve:

Input:

Level 2: {'1/9*X^2+1/16*Y^2-1' '4*Y=2/9*X*ß' 4*X=1/8*Y*ß'}

Level 1: { X Y ß } 20 CF SOLVE

Output

Level 1: { '4*X-3*Y=0' 'Y^2-8=0' 'ß-24=0' }

{ '4*X+3*Y=0' 'Y^2-8=0' 'ß+24=0' }

Level 1: { 'X=s1*(3*Y/4)' 'Y=s1*2.82842712475' 'ß=s1*124' } *s1=±

___________________________________

Example #5: Solve:

Input:

Level 2: { '2*X-3*Y-4*Z-49' '4*X=2*ß' '2*Y=-(3*ß)' '6*Z=-(4*ß)' }

Level 1: { X ß Y Z } SOLVE

Output #1: { 'ß-6=0' 'X-3=0' 'Y+9=0' 'Z+4=0' }

Output #2: { 'ß=6' 'X=3' 'Y=-9' 'Z=-4' }

___________________________________

Example #6: Solve:

Input:

Level 2:{ 'X^2+Y^2-2' 'X+Z-1' '1=2*X*ß+µ' '1=2*Y*ß' '1=µ' }

Level 1:{ X Y Z ß µ } SOLVE

Output #1: { 'X=0' 'Y-4*ß=0' 'Z-1=0' '-µ+1=0' '8*ß^2-1=0' }

Output #2: { 'X=0' 'Y=4*ß' 'Z=1' 'µ=1' 'ß^2=1/8' }

___________________________________

{ 'X^2+Y^2=r^2' '(X-r)^2+Y^2=r^2' }

{ X Y } SOLVE

{{ 'X^2+Y^2=0' 'r=0' }

{ '2*X-r=0' '4*Y^2-3*r^2=0' }} EVAL {X Y} ISOL

{ 'X=r/2' 'Y=s1*ƒ(3*r^2/4)' }

See also: LINSOLVE

_____________________________________________________________

TABVAL similar to TABLE of the (DERIVE V5.02)

Description: For an Algebraic Expression or Equations and a List of Values, returns the results of substituting the values for the Variable in the Algebraic Expression or Equations.

Access: Library 697: CAS48 TABVAL

Input:

Level 3: A Vector { xi } of Equations or an Algebraic expression.

Level 2: A Variable to be evaluated

Level 1: A Vector { xi } of values for which the expression is to be evaluated.

Output:

Level 1: A Matrix of values evaluated.

Example #1: Substitute 1, 2, and 3 into 'X^2+1'

Input:

Level 3: 'X^2+1'

Level 2: X

Level 1: { 1 2 3 } TABVAL [ENTER]

Output:

Level 2:

Input: Output:

Example #2: Substitute 0, ‡, ‡/2, ‡/4, 2*‡ for t

Input:

Level 3: { 'X=2*COS(t)' 'Y=2*SIN(t)' 'Z=t' }

Level 2: t

Level 1: { 0 '‡' '‡/2' '‡/4' '2*‡' }TABVAL [ENTER]

Output:

Level 1:

{{ t 'X=2*COS(t)' 'Y=2*SIN(t)' 'Z=t' }

{ 0 'X=2' 'Y=0' 'Z=0' }

{ '‡' 'X=-2' 'Y=0' 'Z=‡' }

{ '‡/2' 'X=0' 'Y=2' 'Z=1/2*‡' }

{ '‡/4' 'X=ƒ2' 'Y=ƒ2' 'Z=1/4*‡' }

{ '2*‡' 'X=2' 'Y=0' 'Z=2*‡' }}

See also: WTABVAL

Input: Output:

___________________________________

{ 'x^2' 'x^3' 'x^4' }

x

5 TABVAL

{{ x 'x^2' 'x^3' 'x^4' }

{ 1 1 1 1 }

{ 2 4 8 16 }

{ 3 9 27 81 }

{ 4 16 64 256 }

{ 5 25 125 625 }}

___________________________________

'J!'

J

« J 0 4 1 » TABVAL

{{ J 'J!' }

{ 0 1 }

{ 1 1 }

{ 2 2 }

{ 3 6 }

{ 4 24 }}

___________________________________

'k^2'

k

{ 2 3 5 7 11 } TABVAL

{{ k 'k^2' }

{ 2 4 }

{ 3 9 }

{ 5 25 }

{ 7 49 }

{ 11 121 }}

___________________________________

{ 'SIN(z)' 'COS(z)' }

z

« 'z' 0 '‡/4' .2 » TABVAL APPROXIMATE

{{ z 'SIN(z)' 'COS(z)' }

{ 0 0 1 }

{ .2 .198669330795 .98006657843 }

{ .4 .389418342387 .921060994074 }

{ .6 .564642473398 .825335614254 }}

____________________________________________________________

DERIVS

Description: Returns the partial derivative of a function (Algebraic expression), with respect to the specified variable.

Access: Library 697: CAS48 DERIVS

Input:

Level 2: A function or a Vector { xi }.

Level 1: variable.

Output: The derivative, or a vector of the derivatives, of the function or functions.

Example #1: Find the derivative of the following function:

Input:

Level 2: '2*LN(TAN(0.5*ƒX))'

Level 1: 'X' DERIVS

Output:

Level 1: '(TAN(1/2*ƒX)^2+1)/(2*ƒX*TAN(1/2*ƒX))'

Input: Output:

Example #2: Find the derivative of the following function:

Input:

Level 2: { 'X^2*Y+Z^2*Y' 'SIN(X)' }

Level 1: 'X' DERIVS

Output:

Level 1: { '2*X*Y' 'COS(X)' }

___________________________________

'F(X)*G(X)'

X DERIVS

'ˆX(F(X))*G(X)+ˆX(G(X))*F(X)'

___________________________________

'F(X)/G(X)'

X DERIVS

'(ˆX(F(X))*G(X)-ˆX(G(X))*F(X))/G(X)^2'

___________________________________

'F(X)^N'

X DERIVS

'N*F(X)^(N-1)*ˆX(F(X))'

___________________________________

'3*X^3-X+7'

X

2 DERIVN

'18*X'

___________________________________

'X^2'

X

-1 DERIVN

'1/3*X^3'

_____________________________________________________________

INTGRS

Description: Calculates symbolically and Evaluate a definite or indefinite integral.

Access: Library 697: CAS48 INTGRS

Input:

Level 4: An Algebraic Expression

Level 3: A Variable

Level 2: The lower limit. NOVAL for out value or indefinite.

Level 1: The upper limit. NOVAL for out value or indefinite.

Output:

Level 2: The result symbolic.

Level 1: The result of the evaluation if you limit.

Example #1: Evaluate „ 0 3'X^3+3*X' dX

Input:

Level 4: 'X^3+3*X'

Level 3: 'X'

Level 2: 0

Level 1: 3

INTGRS [ENTER]

Output:

Level 2: '1/4*X^4+3/2*X^2'

Level 1: '135/4'

See also: WINTGR

Input: Output:

Example #2: Calculate „'1/(X^2-2)'

Input:

Level 4: '1/(X^2-2)'

Level 3: 'X'

Level 2: NOVAL

Level 1: NOVAL

INTGRS [ENTER]

Output:

Level 1: '-(1/4*ƒ2*LN(X+ƒ2))+1/4*ƒ2*LN(X-ƒ2)'

Input: Output:

_____________________________________________________________

GRADIENT

Description: Returns the GRADIENT of an Algebraic Expression.

compute the general form of the gradient in the Cartesian coordinate system

Access: Library 697: CAS48 GRADIENT

Input:

Level 2: An Algebraic Expression.

Level 1: A list containing the Variables.

Output:

Level 1: The GRADIENT of the Algebraic Expression with respect to the specified Variables.

Example #1: Find the gradient of the following algebraic expression of the spatial variables x, y, and z:

Input:

Level 2: '2*X^2*Y+3*Y^2*Z+Z*X'

Level 1: { X Y Z } GRADIENT [ENTER]

Output:

Level 1: { '4*X*Y+Z' '2*X^2+6*Y*Z' 'X+3*Y^2' }

Input: Output:

Example #2: Find the gradient of the following algebraic expression of the spatial variables x, y, and z:

Input:

Level 2: 'X^2*Y+Z^2*Y'

Level 1: { X Y Z } GRADIENT [ENTER]

Output:

Level 1: { '2*X*Y' 'X^2+Z^2' '2*Y*Z' }

Input: Output:

___________________________________

Example #3:

'x*y^2*z^3'

{ x y z } GRADIENT

{ 'y^2*z^3' '2*x*y*z^3' '3*x*y^2*z^2' }

___________________________________

Example #4:

'c*w+x^2+y^3+z^4'

{ w x y z } GRADIENT

{ c '2*x' '3*y^2' '4*z^3' }

___________________________________

Example #5:

'x*y'

{ x y } GRADIENT

{ y x }

___________________________________

{ y x }

{ x y } POTENTIAL

'x*y'

___________________________________

{ 'y^2*z^3' '2*x*y*z^3' '3*x*y^2*z^2' }

{ x y z } POTENTIAL

'x*y^2*z^3'

See also: WDIV

_____________________________________________________________

DIV

Description: Returns the DIVERGENCE of an Vector Function.

Access:

Input:

Level 2: A Vector Function: Vector { xi }.

Level 1: A list containing the Variables (n<=3)

Output:

Level 1: The DIVERGENCE of the Vectorial Function with respect to the specified variables.

Example #1: Find the divergence of the following Vectorial Function.

Input:

Level 2: { 'X^2*Y' 'X^2*Y' 'Y^2*Z' }

Level 1: { X Y Z } DIV [ENTER]

Output:

Level 1: 'X^2+2*X*Y+Y^2'

See also: WDIV

Input: Output:

Example #2: Find the divergence of the following Vectorial Function

Input:

Level 2: { 'Y*COS(X)' 'X*SIN(Y)' }

Level 1: { X Y } DIV [ENTER]

Output:

Level 1: 'X*COS(Y)-Y*SIN(X)'

See also: WDIV

Input: Output:

_____________________________________________________________

CURL

Type: Command

Description: Returns the ROTATIONAL of an Vector Function.

Access: Library 697: CAS48 CURL

Input:

Level 2: A Vector Function: Vector { xi }.

Level 1: A list containing the Variables (n<=3)

Output:

Level 1: The ROTATIONAL of the Vector Function with respect to the specified variables.

Example #1: Find the rotation of the following Vector Function.

Input:

Level 2: { 'X^2*Y' 'X^2*Y' 'Y^2*Z' }

Level 1: { X Y Z } CURL [ENTER]

Output:

Level 1: { '2*Y*Z' 0 '-X^2+2*X*Y' }

Input: Output:

Example #2: Find the rotation of the following Vector Function

Input:

Level 2: { 'X^3*Y^2*Z' 'X^2*Z' 'X^2*Y' }

Level 1: { X Y Z } CURL [ENTER]

Output:

Level 1: { 0 'X^3*Y^2-2*X*Y' '-(2*X^3*Y*Z)+2*X*Z' }

Input: Output:

See also: WDIV

_____________________________________________________________

LAPLACIAN

'x*y^2*z^3'

{ x y z } LAPLACIAN

'6*x*y^2*z+2*x*z^3'

___________________________________

'X^2+2*X*Y'

{ X Y } LAPLACIAN

2

_____________________________________________________________

MAPRG

Description: Applies a specified program to a list of objects or matrix.

Access: Library 697: CAS48 MAPRG

Input: An object or Vector { xi }of objects.

Output: The new applied object the program.

Example #1: Find the approximate value of ‡ /2, 3*e, and 4cos(2).

Input:

Level 2: { '‡/2' '3*e' '4*COS(2)' }

Level 1: « �NUM » MAPRG

Output:

Level 1: { 1.5707963268 8.15484548538 -1.66458734619 }

Example #2: Evaluates.

Input:

Level 2: {{ 'ƒ4*3_N' 'SIN(‡/2)' 'ˆX(X^2*Y)' }}

Level 1: « EVAL » MAPRG

Output 1: {{ '6_N' 1 'ˆX(X^2)*Y+X^2*ˆX(Y)'}} «EVAL» MAPRG

Output 2: {{ '6_N' 1 'ˆX(X)*2*X^(2-1)*Y' }} «EVAL» MAPRG

Output 3: {{ '6_N' 1 '2*X*Y' }}

_____________________________________________________________

XNUM

Description: Converts an object or a Vector { xi } to approximate numeric format.

Access: Library 697: CAS48 XNUM

Input: An object or Vector { xi } of objects.

Output: The objects in numeric format.

Example #1: Find the approximate value of /2, 3*e, and 4cos(2).

Input:

Level 1: { '‡/2' '3*e' '4*COS(2)' } XNUM

Output:

Level 1: { 1.5707963268 8.15484548538 3.99756330808 }

Example #2:

Input:

{{ 'LN(e)' 'EXP(1)' 'i' 'ƒ2' MAXR '0.5*‡' }} XNUM

Output:

{{ 1 2.71828182846 (0,1) 1.41421356237 9.99999999999E499 1.5707963268 }}

See also: XQ XQ� AXM XEVAL MAPRG

_____________________________________________________________

XEVAL

Description: Evaluates an object or a Vector { xi } of objects.

Access: Library 697: CAS48 XEVAL

Input: An object or Vector { xi } of objects.

Output: The objects evaluated.

Example: Evaluate.

{{'ƒ4*3_N' 'ˆX(X^2*Y)' }} XEVAL

Output 1: {{ '6_N' 'ˆX(X^2)*Y+X^2*ˆX(Y)' }} XEVAL

2: {{ '6_N' 'ˆX(X)*2*X^(2-1)*Y'}} XEVAL

3: {{ '6_N' '2*X*Y' }}

See also: MAP XNUM

_____________________________________________________________

P�COEF

Description: Returns a list of coefficients of a polynomial expression.

Access: Library 697: CAS48 P�COEF

Input: Level 2: The polynomial expression.

Level 1: The variable of polynomial.

Output: A Vector { xi } of the coefficient that the polynomial contains.

Example #1: To find the coefficients of the following polynomial

Input:

Level 2: 'X^2+5*X+6'

Level 1: 'X' P�COEF

Output:

Level 1: { 1 5 6 }

Example #2: To find the coefficients of the following polynomial

Input:

Level 2: 'u*Y^5+ß*Y^3+C'

Level 1: 'Y' P�COEF

Output :

Level 1: { u 0 ß 0 0 C }

Example #3: To find the coefficients of the following polynomial

Input:

Level 2: '2*C*L*R*S^2+C*R^2*S+3*L*S+2*R'

Level 1: 'S' P�COEF

Output :

Level 1: { '2*C*L*R' 'C*R^2+3*L' '2*R' }

Example #4: To find the coefficients of the following polynomial

Input:

Level 2: '2*X^3/(X^2+2)'

Level 1: 'X' P�COEF

Output :

Level 2: { 2 0 0 0 }

Level 1: { 1 0 2 }

See also: C�POL C�FRAT FXND

_____________________________________________________________

C�POL & C�FRAT

Example #1: To find the coefficients of the following polynomial

Input:

Level 2: { 1 5 6 }

Level 1: X C�POL

Output :

Level 1: 'X^2+5*X+6'

Example #2: To find the coefficients of the following polynomial

Input:

Level 3: { 2 0 0 0 }

Level 2: { 1 0 2 }

Level 1: X C�FRAT

Output :

Level 1: '2*X^3/(X^2+2)'

_____________________________________________________________

REORDER

Description: Given a polynomial expression and a variable, reorders the variables in the expression in decreasing order.

Access: Library 697: CAS48 REORDER

Input: Level 2: The polynomial expression.

Level 1: The variable with respect to which the reordering is performed.

Output: The reordered expression.

Example #1: Reordered expression.

Input:

Level 2: '2*R+2*C*L*R*S^2+C*R^2*S+3*L*S'

Level 1: 'S' REORDER

Output:

Level 1: '2*R*C*L*S^2+(R^2*C+3*L)*S+2*R'

Example #2: Reordered expression.

Input: Output:

_____________________________________________________________

Obj�

Description: Object to Stack Command:

Separates an object into its components

Access: Library 697: CAS48 Obj�

Input:

Level 1: An object.

Output:

Level 1: Object separate.

Example #1: To Separate.

Input:

Level 1: «« 'X^Y' DUP »» Obj�  [ENTER]

Output:

Level 5: «

Level 4: 'X^Y'

Level 3: DUP

Level 2: »

Level 1: 4

See also: �Prg

_____________________________________________________________

�Prg

Description: process inverse to OBj�

Access: Library 697: CAS48 �Prg

Input:

Level 1: Objects.

Output:

Level 1: An Object.

Example #1: To join

Input:

Level 5: «

Level 4: 'X^Y'

Level 3: DUP

Level 2: »

Level 1: 4 �Prg [ENTER]

Output:

Level 1: « 'X^Y' DUP »

See also: Obj�

_____________________________________________________________

�Id

Description: Create a name of special variable.

Access: Library 697: CAS48 �Id

Input:

Level 1: A string.

Output:

Level 1: A varible.

Example #1:

Input:

Level 1: "CAS49 Var" �Id [ENTER]

Output:

Level 1: 'CAS48 Var'

Example #2:

Input:

Level 1: "„ @ ˆ" �Id [ENTER]

Output:

Level 1: '„ @ ˆ'

_____________________________________________________________

VER

Description: Returns the Computer Algebra System version number, and date of release.

Access: Library 697: CAS48 VER

Input: No input required.

Output: The version and release date of the Computer Algebra System software.

_____________________________________________________________

HESS

Description: Returns the Hessian matrix, determinant of Hessian matrix, critical points, and the gradient of an expression with respect to the specified variables.

Access: Library 697: CAS48 HESS

Input:

Level 2: Algebraic expression or variable.

Level 1: A Vector { xi } containing the variables.

Output:

Level 4: The gradient with respect to the variables.

Level 3: Critical points: Solution of the level 4:

Level 2: The Hessian matrix

Level 1: Determinant of Hessian matrix

Example #1:

Input:

Level 2: '-X^3+4*X*Y-2*Y^2+1'

Level 1: { X Y } HESS

Output:

Level 4: { '-(3*X^2)+4*Y' '4*X-4*Y' }

Level 3: {{ 'X=0' 'Y=0' }

{ '3*X-4=0' '3*Y-4=0' }}

Level 2: {{ '-(6*X)' 4 }

{ 4 -4 }}

Level 1: :DET=: '24*X-16'

Input: Output:

Example #2:

Input:

'X^2*Y^2'

{ X Y } HESS

Output:

Level 4: { '2*X*Y^2' '2*X^2*Y' }

Level 3: {{ 'X=0' }

{ 'Y=0' }}

Level 2: {{ '2*Y^2' '4*X*Y' }

{ '4*X*Y' '2*X^2' }}

Level 1: :DET=: '-(12*X^2*Y^2)'

Input: Output:

See also: WHESS

_____________________________________________________________

MLAGRANGE

Description: Returns the Multipliers of Lagrange.

Access: Library 697: CAS48 MLAGRANGE

Input:

Level 3: Algebraic expression or variable.

Level 2: A Vector { xi } containing the variables.

Level 1: Restrictions.

Output:

Level 2: The gradient with respect to the variables and restrictions.

Level 1: Solution of the level 2:

Example #1:

Input:

'4*X*Y'

{ 'X^2/9+Y^2/16-1' }

{ X Y } MLAGRANGE

Output:

Level 3: { '4*Y' '4*X' }

Level 2:

{{ '1/9*X^2+1/16*Y^2-1' }

{ '4*Y=2/9*X*ß' }

{ '4*X=1/8*Y*ß' }}

Level 1:

{{ '4*X-3*Y=0' 'Y^2-8=0' 'ß-24=0' }

{ '4*X+3*Y=0' 'Y^2-8=0' 'ß+24=0' }}

Input: Output 2: Output 1:

See also: WLAGRANGE

_____________________________________________________________

JACOBIAN similar to JACOBIAN of the (DERIVE V5.02)

Description: Returns the Jacobian matrix of a parametric function.

simplifies to the Jacobian matrix of the above transformation, where • = { •1 •2 •m } and Jacobian . The Jacobian is the n by m matrix of partial derivatives of u, with ˆui/ˆ•j located at row i and column j. For example, the transformation from parabolic to Cartesian coordinates is x = (w²-v²)/2 and y = w·v and

Access: Library 697: CAS48

Input:

level 2: A vector { xi } representing a parametric function.

level 1: A vector { xi } containing the variables.

Output:

Level 2: The Jacobian matrix

Level 1: DET(#2)

Example #1:

Input:

{ '(w^2-v^2)/2' 'w*v' }

{ w v } JACOBIAN

Output:

Level 2: {{ w '-v' }

{ v w }}

Level 1: :DET=: 'v^2+w^2'

___________________________________

Example #2:

Input:

{ 'r*COS(ß)' 'r*SIN(ß)' ß }

{ r ß } JACOBIAN

Output:

Level 2: {{ I J K }

{ 'COS(ß)' 'SIN(ß)' 0 }

{ '-(r*SIN(ß))' 'r*COS(ß)' 1 }}

Level 1: :DET=: { 'SIN(ß)' '-COS(ß)' r }

See also: WJACOBI

_____________________________________________________________

REF similar to ref() of the (TI92+) or REF of the HP49 (but it accepts symbolic arguments and not squared martix )

Description: subdiagonal reduction. Reduces a matrix to echelon form.

Access: Library 696: LINEAL REF

Flags: Solution of echelon. if (flag 23 SF).

Input:

Level 1: A(m × n) Symbolic/numeric Matrix.

Output:

Level 2: A list of solutions of the echelon. if (flag 23 SF).

Level 1: The equivalent matrix in echelon form.

Example #1: Reduces the following matrix m × m

Input:

Level 1: [[ 2 4 6 ]

[ 4 5 6 ]

[ 3 1 -2 ]] REF [ENTER]

Output:

Level 2: {{ 'R1=Row1/2' }

{ 'R2=Row2+Row1*-4' }

{ 'R3=Row3+Row1*-3' }

{ 'R2=Row2/-3' }

{ 'R3=Row3+Row2*5' }

{ 'R3=Row3/-1' }}

Level 1: [[ 1 2 3 ]

[ 0 1 2 ]

[ 0 0 1 ]]

Input: Output:

A u

___________________________________

Example #2: Reduces the following matrix m × m

Input:

Level 1: [[ (.5,-60) (.5,0) (0,0) ]

[ (0,0) (.5,-60) (.5,0) ]

[ (.5,0) (0,0) (.5,-60) ]] REF [ENTER]

Output:

Level 2:

{{ 'R1=Row1/(1/2-60*i)' }

{ 'R3=Row3+Row1*(-1/2)' }

{ 'R2=Row2/(1/2-60*i)' }

{ 'R3=Row3+Row2*-(-(60/14401*i)-1/28802)' }

{ 'R3=Row3/(-(12443327940/207388801*i)+103687201/207388801)' }}

Level 1: [[ (1,0) (6.94396222485E-5,8.33275466981E-3) (0,0) ]

[ (0,0) (1,0) (6.94396222485E-5,8.33275466981E-3) ]

[ (0,0) (0,0) (1,0) ]]

Input: Output:

A u

___________________________________

Example #3: Reduces the following symbolic matrix m × n

Input:

Level 1: {{ 0 1 -1 2 b1 }

{ 1 -2 3 0 b2 }

{ 1 -1 2 2 b3 }

{ -1 3 -4 2 b4 }} REF [ENTER]

Output:

Level 2: {{ 'Exchange(R1,R2)' }

{ 'R3=Row3+Row1*-1' }

{ 'R4=Row4+Row1*1' }

{ 'R3=Row3+Row2*-1' }

{ 'R4=Row4+Row2*-1' }

{ 'R3=Row3/(-b1-b2+b3)' }

{ 'R4=Row4+Row3*-(-b1+b2+b4)' }}

Level 1: {{ 1 -2 3 0 b2 }

{ 0 1 -1 2 b1 }

{ 0 0 0 0 1 }

{ 0 0 0 0 0 }

Input: Output:

A u

___________________________________

Example #4: {{ a b c }

{ E f g }} REF

{{ 'R1=Row1/a' }

{ 'R2=Row2+Row1*-E' }

{ 'R2=Row2/((-(E*b)+a*f)/a)' }}

{{ 1 'b/a' 'c/a' }

{ 0 1 '(E*c-a*g)/(E*b-a*f)' }}

See also: RREF2 oref

_____________________________________________________________

lu similar to LU() of the (TI92+) or lu of the HP49 (but it accepts symbolic arguments and not squared martix )

Description: lu Decomposition of a Matrix Command.

Returns the lu decomposition of a m × n Symbolic/numeric Matrix.

Access: Library 696: LINEAL lu

Input:

Level 1: A Matrix[[ brackets ]] or Symbolic Matrix {{ xi }}.

Output:

Level 3: P is a permutation matrix. Output:

Level 2: l is an upper-triangular matrix.

Level 1: u is a matrix in echelon.

Where P x A = l × u

Example #1: Factor the following matrix m×m

[[ 0 1 0 0 ]

[ 1 0 0 0 ]

[ 0 0 1 0 ]

[ 0 0 0 1 ]]P

[[ 0 1 -1 2 ]

[ 1 -2 3 0 ]

[ 1 -1 2 2 ]

[ -1 3 -4 2 ]]A

[[ 1 0 0 0 ]

[ 0 1 0 0 ]

[ 1 1 1 0 ]

[ -1 1 0 1 ]]l

[[ 1 -2 3 0 ]

[ 0 1 -1 2 ]

[ 0 0 0 0 ]

[ 0 0 0 0 ]]u Where P x A = l × u

Input: Output:

PA lu

___________________________________

Example #3: Factor the following symbolic matrix 3 × 4

[[ 0 1 0 ]

[ 1 0 0 ]

[ 0 0 1 ]] P

[[ 0 1 -1 2 ]

[ 1 -2 3 0 ]

[ 1 -1 2 2 ]] A

[[ 1 0 0 ]

[ 0 1 0 ]

[ 1 1 1 ]] l

[[ 1 -2 3 0 ]

[ 0 1 -1 2 ]

[ 0 0 0 0 ]] u

___________________________________

Example #2: Factor the following symbolic matrix m × n

[[ 1 0 0 ]

[ 0 1 0 ]

[ 0 0 1 ]] P

[[ 6 12 18 ]

[ 5 14 31 ]

[ 3 8 18 ]] A

[[ 1 0 0 ]

[ .833333333333 1 0 ]

[ .5 .5 1 ]] l

[[ 6 12 18 ]

[ 0 4 16 ]

[ 0 0 1 ]] u

___________________________________

Example #3:

[[ 1 0 ]

[ 0 1 ]] P

{{ m n }

{ q p }} A

{{ 1 0 }

{ 'q/m' 1 }} l

{{ m n }

{ 0 '(m*p-n*q)/m' }} u

See also: LU

_____________________________________________________________

qr similar to QR() of the (TI92+) or qr of the (HP49)

Description: qr Factorization of a Matrix Command: Returns the qr factorization of a m × n numeric Matrix.

Access: Library 696: LINEAL qr

Input:

Level 1: A Matrix [[ xij ]] or Matrix {{ xij }}.

Output:

Level 2: q is an m × m orthogonal matrix.

Level 1: r is an m × n upper trapezoidal matrix.

Where A = q × r

q*TRAN(q)=IDN(m)

Example #1: Factor the following matrix m×m

Input:

Level 1: [[ -.5 -1 1 ]

[ 1 0 .5 ]

[ 0 -1 1 ]] qr [ENTER]

Output:

Level 2: {{ '-(1/5*ƒ5)' '-(4/15*ƒ5)' '2/3' }

{ '2/5*ƒ5' '-(2/15*ƒ5)' '1/3' }

{ 0 '-(1/3*ƒ5)' '-(2/3)' }}

Level 1: {{ '1/2*ƒ5' '1/5*ƒ5' 0 }

{ 0 '3/5*ƒ5' '-(2/3*ƒ5)' }

{ 0 0 '1/6' }} MULT

Input: Output:

A qr

Example #2: Factor the following matrix m×m

Input:

Level 1: {{ m n }

{ k p }} qr [ENTER]

Output:

Level 2: {{ 'm/ƒ(k^2+m^2)' 'k/ƒ(k^2+m^2)' }

{ 'k/ƒ(k^2+m^2)' '-m/ƒ(k^2+m^2)' }}

Level 1: {{ 'ƒ(k^2+m^2)' '(k*p+m*n)/ƒ(k^2+m^2)' }

{ 0 '(k*n-m*p)/ƒ(k^2+m^2)' }}

See also: QR

_____________________________________________________________

PCAR similar to PCAR of the HP49

Description: Returns the characteristic polynomial of a m×m matrix.

Access: Library 696: LINEAL PCAR

Input:

Level 1: A square Matrix [[ xij ]] or square Symbolic Matrix {{ xi }}

Output:

Level 2: The characteristic polynomial of the matrix.

Level 1: A Vector { xi } of the eigenvalues form {Root1 Root2 . . .}

Example #1: Find the characteristic polynomial of the following matrix:

Input:

Level 1: [[ 5 8 16 ]

[ 4 1 8 ]

[ -4 -4 -11 ]] PCAR [ENTER]

Output:

Level 2: 'X^3+5*X^2+3*X-9'

Level 1: { :X1: -3 :X2: -3 :X3: 1 }

Input: Output:

___________________________________

Example #2: Find the characteristic polynomial of the following matrix:

Input:

Level 1: [[ 4 1 ]

[ 0 4 ]] PCAR [ENTER]

Output:

Level 2: 'X^2-8*X+16'

Level 1: { :X1: 4 :X2: 4 }

Input: Output:

___________________________________

Example #3: Find the characteristic polynomial of the following matrix:

Input:

Level 1: [[ .5 .5 0 ]

[ 0 .5 .5 ]

[ .5 0 .5 ]] 'X' PCAR [ENTER]

Output:

Level 2: 'X^3-3/2*X^2+3/4*X-1/4'

Level 1: { :X1: 1 :X2: (.25,.433012701892) :X3: (.25,-.433012701892) }

Input: Output 1:

___________________________________

Example #4:

_____________________________________________________________

{{ 2 X }

{ 1 3 }} ADJOINT

{{ 3 '-X' }

{ -1 2 }}

_____________________________________________________________

Polynomial Function

FXND similar to comDenom() of the (TI92+) or FXND of the HP49

returns a reduced ratio of a fully expanded numerator and a fully

expanded denominator.

'(Y^2+Y)/(1+X)^2+Y^2+Y' FXND

numerator: 'X^2*Y^2+X^2*Y+2*X*Y^2+2*X*Y+2*Y^2+2*Y'

denominator: 'X^2+2*X+1'

___________________________________

'(X^2+X+1)/(X+1)+(Y^2+Y+1)/(Y+1)=0' FXND

numerator: 'X^2*Y+X^2+X*Y^2+2*X*Y+2*X+Y^2+2*Y+2'

denominator: 'X*Y+X+Y+1'

___________________________________

3.58333333333 FXND

43

12

_____________________________________________________________

MULT2POL similar to conv of MATLAB VERSION 6.0

[ 1 2 3 ]

[ 4 5 6 ]

'X' MULT2POL

'4*X^4+13*X^3+28*X^2+27*X+18'

'X' P�COEF AXL

[ 4 13 28 27 18 ]

___________________________________

[ 1 2 3 ]

{ 4 5 k }

'X' MULT2POL

'4*X^4+13*X^3+(22+k)*X^2+(15+2*k)*X+3*k' FACTOR

'(X^2+2*X+3)*(4*X^2+5*X+k)'

___________________________________

'X^2+2*X+3'

'4*X^2+5*X+k'

'X' MULT2POL

'4*X^4+13*X^3+(22+k)*X^2+(15+2*k)*X+3*k'

'k=6' SUBST SIMPLIFY

'4*X^4+13*X^3+28*X^2+27*X+18'

_____________________________________________________________

DIV2POL similar to propFrac() of the (TI89) or DIV2 of the HP49 or deconv of MATLAB VERSION 6.0

N/D=Q+R/D

'2*X^3'

'X^2+2'

'X' DIV2POL

quotient: '2*X'

remainder: '-(4*X)'

___________________________________

[ 4 13 28 27 18 ]

[ 1 2 3 ]

'X' DIV2POL

quotient: '4*X^2+5*X+6'

remainder: 0

___________________________________

'5*X^4+4*X^3+3*X^2+2*X+1'

'X+i'

'X' DIV2POL

quotient: '5*X^3+(4-5*i)*X^2+(-2-4*i)*X+(-2,2)'

remainder: '3+2*i'

___________________________________

'X^3+(-1-A-B)*X^2+(A*B+A+B)*X-A*B'

'X-(1+C/3)'

'X' DIV2POL

quotient:

'X^2+(-1+1/3*(3+C)-A-B)*X+(1/3*(-1+1/3*(3+C)-A-B)*(3+C)+A*B+A+B)'

remainder:

'1/3*(1/3*(-1+1/3*(3+C)-A-B)*(3+C)+A*B+A+B)*(3+C)-A*B' FACTOR

'1/27*C*(3*B-C-3)*(3*A-C-3)'

_____________________________________________________________

PROPFRAC similar to propFrac() of the (TI92+) or PROPFRACT of the HP49

Q+R/D

'2*X^3/(X^2+2)'

'X' PROPFRAC

'2*X-4*X/(X^2+2)'

_________________________________

'43/12'

'X' PROPFRAC

'3+7/12'

___________________________________

'(X^2+X+1)/(X+1)+(Y^2+Y+1)/(Y+1)'

'X' PROPFRAC

'(X*Y+X+Y^2+Y+1)/(Y+1)+(Y+1)/(X*Y+X+Y+1)'

_____________________________________________________________

GCD similar to GCD of the HP49

'X^2+2*X+1'

'X^2-1' GCD

'X+1'

___________________________________

18

33 GCD

3

_____________________________________________________________

LCM similar to LCM of the HP49

'X^2+2*X+1'

'X^2-1' LCM

'X^3+X^2-X-1'

___________________________________

18

33 LCM

198

_____________________________________________________________

POLEVALM similar to polyvalm of MATLAB VERSION 6.0

[[ 2 4 5 ]

[ -1 0 3 ]

[ 7 1 5 ]]

[ 1 0 -2 -5 ]

X POLEVALM

[[ 377 179 439 ]

[ 111 81 136 ]

[ 490 253 639 ]]

___________________________________

{{ 2 4 5 }

{ -1 0 3 }

{ 7 1 a }}

[ 1 0 -2 -5 ]

X POLEVALM

{{ '35*a+202' '5*a+154' '5*a^2+22*a+204' }

{ '21*a+6' '3*a+66' '3*a^2-5*a+86' }

{ '7*a^2+13*a+250' 'a^2+28*a+88' 'a^3+74*a+144' }}

_____________________________________________________________

'1.3-3.2*i' CEILING

'2-3*i'

_____________________________________________________________

LCXM similar to LCXM of the HP49

2

3

« � I J 'I+2*J' » LCXM

[[ 3 5 7 ]

[ 4 6 8 ]]

___________________________________

3

3

« � X Y '1/(X+Y-1)' » LCXM EXACT

{{ 1 '1/2' '1/3' }

{ '1/2' '1/3' '1/4' }

{ '1/3' '1/4' '1/5' }}

_____________________________________________________________

3 HILBERT

{{ 1 '1/2' '1/3' }

{ '1/2' '1/3' '1/4' }

{ '1/3' '1/4' '1/5' }}

_____________________________________________________________

'X^2+1'

1

X HORNER

quotient (P/X-a): 'X+1'

P(a): 2

___________________________________

'5*X^4+4*X^3+3*X^2+2*X+1'

'-i'

X HORNER

quotient (P/X-a): '5*X^3+(4,-5)*X^2+(-2,-4)*X+(-2,2)'

P(a): '3+2*i'

___________________________________

'EXP(X)'

X

s LAPLACE

'1/(s-1)'

___________________________________

't^2'

t

S LAPLACE

'2/S^3'

___________________________________

't^2*EXP(5*t)'

t

S LAPLACE

'2/(S^3-15*S^2+75*S-125)' FACTOR

'INV(1/2*(S-5)^3)'

___________________________________

'X*COS(a*X+b)'

X

S LAPLACE

'(COS(b)*S^2-2*SIN(b)*a*S-COS(b)*a^2)/(S^4+2*a^2*S^2+a^4)'

S P�COEF

{ 'COS(b)' '-(2*SIN(b)*a)' '-(COS(b)*a^2)' }

{ 1 0 '2*a^2' 0 'a^4' }

___________________________________

'1/(s-1)'

s

X ILAPLACE

'e^X'

___________________________________

'2/S^3'

S

t ILAPLACE

't^2'

___________________________________

'INV(1/2*(S-5)^3)'

S

t ILAPLACE

't^2*e^(5*t)'

___________________________________

'1/(S^4+6*S^3+14*S^2+16*S+8)'

S

t ILAPLACE EXACT

'1/2*e^-(2*t)+1/2*t*e^-(2*t)-1/2*EXP(1)^-t*COS(t)'

'(X^4+3.414*X^2+1)/(2.613*X^3+2.613*X)'

X PROPFRAC2

'.382701875239*X+1/(1.08243579122*X+1/(1.57719616481*X+1

/(1.53056420907*X)))'

___________________________

[[ 2 4 7 ]

[ 1 2 3 ]

[ 1 5 3 ]]

{ 3 2 } MINOR

{{ 2 7 }

{ 1 3 }}

___________________________

[[ 2 4 7 ]

[ 1 2 3 ]

[ 1 5 3 ]]

{ 3 2 } COFACT

1

___________________________

{{ a b c }

{ d E f }

{ g h I }}

{ 3 1 } MINOR

{{ b c }

{ E f }}

___________________________

{{ 3 2 1 }

{ 1 -1 3 }

{ 5 4 -2 }}

{{ 7 }

{ 3 }

{ 1 }}

{ X Y Z } CRAMER

{ :X: -3 :Y: 6 :Z: 4 }

___________________________

Input(Levels A b X):

3: {{ a b }

{ c d }}

2: {{ 1 }

{ 2 }}

1: {{ x1 }

{ x2 }} CRAMER [ENTER]

Output(Levels):

1:

{ :x1: '(-(2*b)+d)/(a*d-b*c)'

:x2: '(2*a-c)/(a*d-b*c)' }

___________________________

3 RANDSYSMAT

{{ -5 -1 '-(3/2)' }

{ -1 1 -2 }

{ '-(3/2)' -2 -9 }}

___________________________

3 RANDASYSMAT

{{ 0 9/2' '9/2' }

{ '-(9/2)' 0 '9/2' }

{ '-(9/2)' '-(9/2)' 0 }}

___________________________

{ 3 4 } RANDUPPERMAT

{{ 9 -9 -8 9 }

{ 0 -2 '-65/9' 2 }

{ 0 0 '-53/3' 6 }}

___________________________

{ 3 4 } RANDLOWERMAT

{{ 7 0 0 0 }

{ -6 '23/7' 0 0 }

{ -1 '5/7' '-36/23' 0 }}

___________________________

{ 2 4 } RANDSTOCMAT

{{ '4/31' '5/14' }

{ '7/31' '3/7' }

{ '11/31' '1/7' }

{ '9/31' '1/14' }}

SUMA(COL)=1

___________________________

[[ .5 .5 0 ]

[ 0 .5 .5 ]

[ .5 0 .5 ]]

'Y'PMINI

'Y^3-3/2*Y^2+3/4*Y-1/4'

___________________________

[[ 1 2 3 ]

[ 4 5 6 ]] IMAGE

[[ 1 0 ]

[ 0 1 ]]

___________________________

[[ 1 2 3 ]

[ 4 5 6 ]] KER

[[ -1 2 -1 ]]

___________________________

[[ 1 2 3 ]

[ 4 5 6 ]

[ 7 8 9 ]]

[[ 1 2 3 ]

[ 4 5 6 ]

[ 7 5 6 ]] EGBS

[[ 1 -1 0 ]

[ 1 0 -1 ]]

___________________________

{ 1 3 5 7 9 }

{ 2 3 5 8 13 } LINTERSECTION or LAND

{ 3 5 }

___________________________

DEL ELEMENTS OF 2 IN LIST 1

{ 1 2 3 4 5 6 7 8 9 10 }

{ 2 3 5 7 } LDIFFERENCE

{ 1 4 6 8 9 10 }

___________________________

{ 1 3 5 7 9 }

{ 2 3 5 8 13 } LOR or LUNION

{ 1 3 5 7 9 2 8 13 }

___________________________

{ 1 3 5 7 9 }

{ 2 3 5 8 13 } LXOR

{ 1 7 9 2 8 13 }

___________________________

{ 1 3 5 7 9 } 1 3 LSUB

{ 7 9 }

___________________________

{ 'COS(X)' 'EXP(X)' 'pi' 'SQ(Y)' }

{ 'pi' 'i' 'LN(X)' 'EXP(X)' } LAND

{ 'EXP(X)' 'pi' }

___________________________

{ 'COS(X)' 'EXP(X)' 'pi' 'SQ(Y)' }

{ 'pi' 'i' 'LN(X)' 'EXP(X)' } LDIFFERENCE

{ 'COS(X)' 'SQ(Y)' }

___________________________

{ 'COS(X)' 'EXP(X)' 'pi' 'SQ(Y)' }

{ 'pi' 'i' 'LN(X)' 'EXP(X)' } LOR

{ 'COS(X)' 'EXP(X)' 'pi' 'SQ(Y)' 'i' 'LN(X)' }

___________________________

{ 'COS(X)' 'EXP(X)' 'pi' 'SQ(Y)' }

{ 'pi' 'i' 'LN(X)' 'EXP(X)' } LXOR

{ 'COS(X)' 'SQ(Y)' 'i' 'LN(X)' }

___________________________

{ 1 1 0 0 }

{ 1 0 1 1 } LAND

{ 1 1 0 }

___________________________

{ 1 1 0 0 }

{ 1 0 1 1 } LOR

{ 1 1 0 0 1 }

___________________________

{ 1 1 0 0 }

{ 1 0 1 1 } LDIFFERENCE

{ 0 }

___________________________

{ 1 1 0 0 }

{ 1 0 1 1 } LXOR

{ 0 1 }

___________________________

{ 1 1 0 0 } LDELMULT

{ 1 0 }

___________________________

{ 1 0 1 1 }

1

2 LSWAP

{ 0 1 1 1 }

___________________________

{ 1 0 1 1 }

{ 1 0 1 1 } LSAME

1

___________________________

{ X « X ISPRIME? » }

X 1 16 1 VECTOR

[[ 1 0 ]

[ 2 1 ]

[ 3 1 ]

[ 4 0 ]

[ 5 1 ]

[ 6 0 ]

[ 7 1 ]

[ 8 0 ]

[ 9 0 ]

[ 10 0 ]

[ 11 1 ]

[ 12 0 ]

[ 13 1 ]

[ 14 0 ]

[ 15 0 ]

[ 16 0 ]]

___________________________

{ X « X DIVISORS » }

X 1 16 1 VECTOR

{

{ 1 { 1 }}

{ 2 { 1 2 }}

{ 3 { 1 3 }}

{ 4 { 1 2 4 }}

{ 5 { 1 5 }}

{ 6 { 1 2 3 6 }}

{ 7 { 1 7 }}

{ 8 { 1 2 4 8 }}

{ 9 { 1 3 9 }}

{ 10 { 1 2 5 10 }

{ 11 { 1 11 }}

{ 12 { 1 2 3 4 6 12 }}

{ 13 { 1 13 }}

{ 14 { 1 2 7 14 }}

{ 15 { 1 3 5 15 } }

{ 16 { 1 2 4 8 16 }}

}

___________________________

{ X « X PRIMEDIVIS » }

X 1 16 1 VECTOR

{

{ 1 { } }

{ 2 { 2 } }

{ 3 { 3 } }

{ 4 { 2 } }

{ 5 { 5 } }

{ 6 { 2 3 } }

{ 7 { 7 } }

{ 8 { 2 } }

{ 9 { 3 } }

{ 10 { 2 5 } }

{ 11 { 11 }}

{ 12 { 2 3 }}

{ 13 { 13 }}

{ 14 { 2 7 }}

{ 15 { 3 5 } }

{ 16 { 2 }}

}

___________________________

{ X « X PERFEC? » }

X 6 30 1 VECTOR

[[ 6 1 ]

[ 7 0 ]

[ 8 0 ]

[ 9 0 ]

[ 10 0 ]

[ 11 0 ]

[ 12 0 ]

[ 13 0 ]

[ 14 0 ]

[ 15 0 ]

[ 16 0 ]

[ 17 0 ]

[ 18 0 ]

[ 19 0 ]

[ 20 0 ]

[ 21 0 ]

[ 22 0 ]

[ 23 0 ]

[ 24 0 ]

[ 25 0 ]

[ 26 0 ]

[ 27 0 ]

[ 28 1 ]

[ 29 0 ]

[ 30 0 ]]

___________________________

{ X « X FACTOR » }

X -1 20 1 VECTOR

{{ -1 -1 }

{ 0 0 }

{ 1 1 }

{ 2 2 }

{ 3 3 }

{ 4 '2^2' }

{ 5 5 }

{ 6 '2*3' }

{ 7 7 }

{ 8 '2^3' }

{ 9 '3^2' }

{ 10 '2*5' }

{ 11 11 }

{ 12 '2^2*3' }

{ 13 13 }

{ 14 '2*7' }

{ 15 '3*5' }

{ 16 '2^4' }

{ 17 17 }

{ 18 '2*3^2' }

{ 19 19 }

{ 20 '2^2*5' }}

___________________________

{ X « X FACTORS » }

X -1 20 1 VECTOR

{{-1 {{ -1 1 }} }

{ 0 {{ 0 1 }} }

{ 1 {{ 1 1 }} }

{ 2 {{ 2 1 }} }

{ 3 {{ 3 1 }} }

{ 4 {{ 2 2 }} }

{ 5 {{ 5 1 }} }

{ 6 {{ 2 1 } { 3 1 }} }

{ 7 {{ 7 1 }} }

{ 8 {{ 2 3 }} }

{ 9 {{ 3 2 }} }

{ 10 {{ 2 1 } { 5 1 } } }

{ 11 {{ 11 1 }} }

{ 12 {{ 2 2 } { 3 1 }} }

{ 13 {{ 13 1 }} }

{ 14 {{ 2 1 } { 7 1 }} }

{ 15 {{ 3 1 } { 5 1 }} }

{ 16 {{ 2 4 }} }

{ 17 {{ 17 1 }} }

{ 18 {{ 2 1 } { 3 2 }} }

{ 19 {{ 19 1 }} }

{ 20 {{ 2 2 } { 5 1 } }}

}

___________________________

{{ 5 '8/2' '4/2' }

{ '8/2' 5 2 }

{ '4/2' 2 2 }}

{ X Y Z } AXQ

'5*X^2+8*X*Y+4*X*Z+5*Y^2+4*Y*Z+2*Z^2=100'

___________________________

{{ 5 '-3/2' 4 '-1/2' }

{ '-3/2' 4 '9/2' '7/2' }

{ 4 '9/2' 2 3 }

{ '-1/2' '7/2' 3 9 }}

DIMMAT 1 GET MAKEVX AXQ

'5*W^2-3*W*X+8*W*Y-W*Z+4*X^2+9*X*Y+7*X*Z+2*Y^2+6*Y*Z+9*Z^2'

___________________________

13300

FACTORS MULT->1LIST

{ 2 2 5 2 7 1 19 1 }

___________________________

EJEMPLO DE MATLABV6.0 SISTEMA

[[ 1 1 ]

[ 1 .7408 ]

[ 1 .4493 ]

[ 1 .3329 ]

[ 1 .2019 ]

[ 1 .1003 ]] A(6x2)

[[ .82 ]

[ .72 ]

[ .63 ]

[ .6 ]

[ .55 ]

[ .5 ]] b(6x1)

{ X Y } x(2x1) LINSOLVE

:X: .475943892709

:Y: .341333938746

* SOL-> (TRAN(A)*A)*X=(TRAN(A)*b)

___________________________

6 PASCAL

[[ 1 1 1 1 1 1 ]

[ 1 2 3 4 5 6 ]

[ 1 3 6 10 15 21 ]

[ 1 4 10 20 35 56 ]

[ 1 5 15 35 70 126 ]

[ 1 6 21 56 126 252 ]]

CHOLESKY

[[ 1 1 1 1 1 1 ]

[ 0 1 2 3 4 5 ]

[ 0 0 1 3 6 10 ]

[ 0 0 0 1 4 10 ]

[ 0 0 0 0 1 5 ]

[ 0 0 0 0 0 1 ]]

___________________________

6 MAKEVX MVANDERMONDE

{{ 1 1 1 1 1 1 }

{ U V W X Y Z }

{ 'U^2' 'V^2' 'W^2' 'X^2' 'Y^2' 'Z^2' }

{ 'U^3' 'V^3' 'W^3' 'X^3' 'Y^3' 'Z^3' }

{ 'U^4' 'V^4' 'W^4' 'X^4' 'Y^4' 'Z^4' }

{ 'U^5' 'V^5' 'W^5' 'X^5' 'Y^5' 'Z^5' }}

___________________________

'1/(X*(X+1))' X 'Ÿ' 1 INTGMAX

.693147180555

'1/(X^2+4)' X 'Ÿ' 0 INTGMAX

.785398163398 XQ '1/4*‡'

'1/(EXP(X)+EXP(-X))' X 'Ÿ' 'Ÿ' INTGMAX

'‡/2'

'EXP(2)^X' X 0 'Ÿ' INTGMAX

0.5

___________________________

'X^9+(-191/10-16*i)*X^8+(518/5,2651/10)*X^7+(291/10,-21431/10)*X^6+(-123880898/28569,10145)*X^5+(256059/10,-129071/5)*X^4+(-66412,65216915/1917)*X^3+(868493/10,-215983/10)*X^2+(-279806/5,15604397/3252)*X+(70698/5,1/4*ƒ1880189)'

{ (2,8) (3,-5) (3.1,5) (3,4) (2,0) (1,0) (1,0) (1,0) (3,4) }

___________________________

'Y^4+Y^3-Y-1'

'Y(X)' DESOLVE1

'c1*e^-X+c2*e^X+EXP(-(1/2*X))*(c3*COS(1/2*ƒ3*X)+c4*SIN(1/2*ƒ3*X))'

'Y^7+8*Y^6+28*Y^5+56*Y^4+69*Y^3+52*Y^2+22*Y+4'

'Y(X)' DESOLVE1

'c1*e^-(2*X)+c2*e^-X+c3*(X*e^-X)+c4*(X^2*e^-X)+c5*(X^3*e^-X)+EXP(-X)*(c6*COS(X)+c7*SIN(X))'

___________________________________

Package cashp48.zip	AUTHOR: Jaime Fernando Meza Meza E-MAIL: jaimeza@hotmail.com Visit our Web site at: http://www.unalmed.edu.co/~ameza or http://www.geocities.com/hp4x/ Software and information about the HP49,HP48 at: http://www.hpcalc.org/
Version: 3.3 revision 1 Update: oct/20/2001

Please excuse all the linguistic errors in this Pocket.

English is not my native language.