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	FriCAS Algebra
OrderedVariableList ←	↑	→ RealNumbers

QuadraticForm

Edit detail for QuadraticForm revision 4 of 4

	1 2 3 4
Editor: test1 Time: 2017/01/27 18:18:43 GMT+0
Note:

changed:
-\begin{spad}
-)abbrev domain QFORM QuadraticForm
-QuadraticForm(n, K): T == Impl where
-    n: PositiveInteger
-    K: Field
-    SM ==> SquareMatrix
-    V  ==> DirectProduct
-
-    T ==> AbelianGroup with
-        quadraticForm: SM(n, K) -> %
-            ++ quadraticForm(m) creates a quadratic form from a symmetric,
-            ++ square matrix m.
-        matrix: % -> SM(n, K)
-            ++ matrix(qf) creates a square matrix from the quadratic form qf.
-        elt: (%, V(n, K)) -> K
-            ++ elt(qf,v) evaluates the quadratic form qf on the vector v,
-            ++ producing a scalar.
-        coerce: % -> InputForm
-
-    Impl ==> SM(n,K) add
-        Rep := SM(n,K)
-
-        import List InputForm
-
-        coerce(q:%):InputForm ==
-            convert(['quadraticForm,algCoerceInteractive(q,SM(n,K),
-                InputForm)$Lisp pretend InputForm])
-
-        quadraticForm m ==
-            not symmetric? m =>
-                error "quadraticForm requires a symmetric matrix"
-            m::%
-        matrix q == q pretend SM(n,K)
-        elt(q,v) == dot(v, (matrix q * v))
-\end{spad}




































added:

fricas

(1) -> )show QuadraticForm

 QuadraticForm(n: PositiveInteger,K: Field) is a domain constructor
 Abbreviation for QuadraticForm is QFORM 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------

 ?*? : (Integer, %) -> %               ?*? : (PositiveInteger, %) -> %
 ?+? : (%, %) -> %                     ?-? : (%, %) -> %
 -? : % -> %                           ?=? : (%, %) -> Boolean
 0 : () -> %                           coerce : % -> OutputForm
 latex : % -> String                   matrix : % -> SquareMatrix(n,K)
 opposite? : (%, %) -> Boolean         sample : () -> %
 zero? : % -> Boolean                  ?~=? : (%, %) -> Boolean
 ?*? : (NonNegativeInteger, %) -> %
 convert : % -> InputForm if SquareMatrix(n,K) has KONVERT(INFORM)
 elt : (%, DirectProduct(n,K)) -> K
 quadraticForm : SquareMatrix(n,K) -> %
 subtractIfCan : (%, %) -> Union(%,"failed")

fricas

parse(s:String):InputForm == ncParseFromString(s)$Lisp pretend InputForm

   Function declaration parse : String -> InputForm has been added to 
      workspace.

Type: Void

fricas

parse("Integer")

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Compiling function parse with type String -> InputForm

$\label{eq1}\hbox{\axiomType{Integer}\ }$

(1)

Type: InputForm

Test

fricas

qf := quadraticForm matrix [[1,2],[2,-1]]

$\label{eq2}\left[ \begin{array}{cc} 1 & 2 \ 2 & - 1$

(2)

Type: QuadraticForm(2,Fraction(Integer))

fricas

qf::InputForm

$\label{eq3}\left({quadraticForm \ {\left({squareMatrix \ {\left({matrix \ {\left({ \begin{array}{@{}l} \displaystyle construct \ \cdot \ \ \displaystyle {\left({ \begin{array}{@{}l} \displaystyle construct \ \cdot \ \ \displaystyle 1 \ \cdot \ \ \displaystyle 2$

(3)

Type: InputForm

fricas

unparse %

$\label{eq4}\verb#"quadraticForm(squareMatrix(matrix([[1,2],[2,-1]])))"#$

(4)

Type: String

fricas

parse %

$\label{eq5}\left({quadraticForm \ {\left({squareMatrix \ {\left({matrix \ {\left({ \begin{array}{@{}l} \displaystyle construct \ \cdot \ \ \displaystyle {\left({ \begin{array}{@{}l} \displaystyle construct \ \cdot \ \ \displaystyle 1 \ \cdot \ \ \displaystyle 2$

(5)

Type: InputForm

fricas

interpret(%)$INFORM1(QuadraticForm(2,Fraction Integer))

$\label{eq6}\left[ \begin{array}{cc} 1 & 2 \ 2 & - 1$

(6)

Type: QuadraticForm(2,Fraction(Integer))