TESTING/EIG/dspt21.f(3) Library Functions Manual TESTING/EIG/dspt21.f(3) NAME TESTING/EIG/dspt21.f SYNOPSIS Functions/Subroutines subroutine dspt21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, result) DSPT21 Function/Subroutine Documentation subroutine dspt21 (integer itype, character uplo, integer n, integer kband, double precision, dimension( * ) ap, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( * ) vp, double precision, dimension( * ) tau, double precision, dimension( * ) work, double precision, dimension( 2 ) result) DSPT21 Purpose: DSPT21 generally checks a decomposition of the form A = U S U**T where **T means transpose, A is symmetric (stored in packed format), 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 the 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 ) Packed storage means that, for example, if UPLO='U', then the columns of the upper triangle of A are stored one after another, so that A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if UPLO='L', then the columns of the lower triangle of A are stored one after another in AP, so that A(j+1,j+1) immediately follows A(n,j) in the array AP. This means that A(i,j) is stored in: AP( i + j*(j-1)/2 ) if UPLO='U' AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L' The array VP bears the same relation to the matrix V that A does to AP. For ITYPE > 1, the transformation U is expressed as a product of Householder transformations: If UPLO='U', then V = H(n-1)...H(1), where H(j) = I - tau(j) v(j) v(j)**T and the first j-1 elements of v(j) are stored in V(1:j-1,j+1), (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ), the j-th element is 1, and the last n-j elements are 0. If UPLO='L', then V = H(1)...H(n-1), where H(j) = I - tau(j) v(j) v(j)**T and the first j elements of v(j) are 0, the (j+1)-st is 1, and the (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e., in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .) Parameters 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', AP and VP are considered to contain the upper triangle of A and V. If UPLO='L', AP and VP are considered to contain the lower triangle of A and V. N N is INTEGER The size of the matrix. If it is zero, DSPT21 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. AP AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The original (unfactored) matrix. It is assumed to be symmetric, and contains the columns of just the upper triangle (UPLO='U') or only the lower triangle (UPLO='L'), packed one after another. D D is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri-) diagonal matrix. E E is DOUBLE PRECISION 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 DOUBLE PRECISION 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. VP VP is DOUBLE PRECISION array, dimension (N*(N+1)/2) If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the orthogonal matrix in the decomposition, as described in purpose. *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. TAU TAU is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N**2+N) Workspace. RESULT RESULT is DOUBLE PRECISION 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. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 219 of file dspt21.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 TESTING/EIG/dspt21.f(3)