SSYT21 generally checks a decomposition of the form


    A = U S U**T


 where **T means transpose, A is symmetric, U is orthogonal, and S is


 diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).


 If ITYPE=1, then U is represented as a dense matrix; otherwise U is


 expressed as a product of Householder transformations, whose vectors


 are stored in the array 'V' and whose scaling constants are in 'TAU'.


 We shall use the letter 'V' to refer to the product of Householder


 transformations (which should be equal to U).


 Specifically, if ITYPE=1, then:


    RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and


    RESULT(2) = | I - U U**T | / ( n ulp )


 If ITYPE=2, then:


    RESULT(1) = | A - V S V**T | / ( |A| n ulp )


 If ITYPE=3, then:


    RESULT(1) = | I - V U**T | / ( n ulp )


 For ITYPE > 1, the transformation U is expressed as a product


 V = H(1)...H(n-2),  where H(j) = I  -  tau(j) v(j) v(j)**T and each


 vector v(j) has its first j elements 0 and the remaining n-j elements


 stored in V(j+1:n,j).

ITYPE

          ITYPE is INTEGER


          Specifies the type of tests to be performed.


          1: U expressed as a dense orthogonal matrix:


             RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and


             RESULT(2) = | I - U U**T | / ( n ulp )


          2: U expressed as a product V of Housholder transformations:


             RESULT(1) = | A - V S V**T | / ( |A| n ulp )


          3: U expressed both as a dense orthogonal matrix and


             as a product of Housholder transformations:


             RESULT(1) = | I - V U**T | / ( n ulp )

UPLO

          UPLO is CHARACTER


          If UPLO='U', the upper triangle of A and V will be used and


          the (strictly) lower triangle will not be referenced.


          If UPLO='L', the lower triangle of A and V will be used and


          the (strictly) upper triangle will not be referenced.

N

          N is INTEGER


          The size of the matrix.  If it is zero, SSYT21 does nothing.


          It must be at least zero.

KBAND

          KBAND is INTEGER


          The bandwidth of the matrix.  It may only be zero or one.


          If zero, then S is diagonal, and E is not referenced.  If


          one, then S is symmetric tri-diagonal.

A

          A is REAL array, dimension (LDA, N)


          The original (unfactored) matrix.  It is assumed to be


          symmetric, and only the upper (UPLO='U') or only the lower


          (UPLO='L') will be referenced.

LDA

          LDA is INTEGER


          The leading dimension of A.  It must be at least 1


          and at least N.

D

          D is REAL array, dimension (N)


          The diagonal of the (symmetric tri-) diagonal matrix.

E

          E is REAL array, dimension (N-1)


          The off-diagonal of the (symmetric tri-) diagonal matrix.


          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and


          (3,2) element, etc.


          Not referenced if KBAND=0.

U

          U is REAL array, dimension (LDU, N)


          If ITYPE=1 or 3, this contains the orthogonal matrix in


          the decomposition, expressed as a dense matrix.  If ITYPE=2,


          then it is not referenced.

LDU

          LDU is INTEGER


          The leading dimension of U.  LDU must be at least N and


          at least 1.

V

          V is REAL array, dimension (LDV, N)


          If ITYPE=2 or 3, the columns of this array contain the


          Householder vectors used to describe the orthogonal matrix


          in the decomposition.  If UPLO='L', then the vectors are in


          the lower triangle, if UPLO='U', then in the upper


          triangle.


          *NOTE* If ITYPE=2 or 3, V is modified and restored.  The


          subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')


          is set to one, and later reset to its original value, during


          the course of the calculation.


          If ITYPE=1, then it is neither referenced nor modified.

LDV

          LDV is INTEGER


          The leading dimension of V.  LDV must be at least N and


          at least 1.

TAU

          TAU is REAL array, dimension (N)


          If ITYPE >= 2, then TAU(j) is the scalar factor of


          v(j) v(j)**T in the Householder transformation H(j) of


          the product  U = H(1)...H(n-2)


          If ITYPE < 2, then TAU is not referenced.

WORK

          WORK is REAL array, dimension (2*N**2)

RESULT

          RESULT is REAL array, dimension (2)


          The values computed by the two tests described above.  The


          values are currently limited to 1/ulp, to avoid overflow.


          RESULT(1) is always modified.  RESULT(2) is modified only


          if ITYPE=1.

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