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*DECK TRED1
SUBROUTINE TRED1 (NM, N, A, D, E, E2)
C***BEGIN PROLOGUE TRED1
C***PURPOSE Reduce a real symmetric matrix to symmetric tridiagonal
C matrix using orthogonal similarity transformations.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C1B1
C***TYPE SINGLE PRECISION (TRED1-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure TRED1,
C NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C This subroutine reduces a REAL SYMMETRIC matrix
C to a symmetric tridiagonal matrix using
C orthogonal similarity transformations.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameter, A, as declared in the calling program
C dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix A. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C A contains the real symmetric input matrix. Only the lower
C triangle of the matrix need be supplied. A is a two-
C dimensional REAL array, dimensioned A(NM,N).
C
C On Output
C
C A contains information about the orthogonal transformations
C used in the reduction in its strict lower triangle. The
C full upper triangle of A is unaltered.
C
C D contains the diagonal elements of the symmetric tridiagonal
C matrix. D is a one-dimensional REAL array, dimensioned D(N).
C
C E contains the subdiagonal elements of the symmetric
C tridiagonal matrix in its last N-1 positions. E(1) is set
C to zero. E is a one-dimensional REAL array, dimensioned
C E(N).
C
C E2 contains the squares of the corresponding elements of E.
C E2 may coincide with E if the squares are not needed.
C E2 is a one-dimensional REAL array, dimensioned E2(N).
C
C Questions and comments should be directed to B. S. Garbow,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED (NONE)
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE TRED1
C
INTEGER I,J,K,L,N,II,NM,JP1
REAL A(NM,*),D(*),E(*),E2(*)
REAL F,G,H,SCALE
C
C***FIRST EXECUTABLE STATEMENT TRED1
DO 100 I = 1, N
100 D(I) = A(I,I)
C .......... FOR I=N STEP -1 UNTIL 1 DO -- ..........
DO 300 II = 1, N
I = N + 1 - II
L = I - 1
H = 0.0E0
SCALE = 0.0E0
IF (L .LT. 1) GO TO 130
C .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
DO 120 K = 1, L
120 SCALE = SCALE + ABS(A(I,K))
C
IF (SCALE .NE. 0.0E0) GO TO 140
130 E(I) = 0.0E0
E2(I) = 0.0E0
GO TO 290
C
140 DO 150 K = 1, L
A(I,K) = A(I,K) / SCALE
H = H + A(I,K) * A(I,K)
150 CONTINUE
C
E2(I) = SCALE * SCALE * H
F = A(I,L)
G = -SIGN(SQRT(H),F)
E(I) = SCALE * G
H = H - F * G
A(I,L) = F - G
IF (L .EQ. 1) GO TO 270
F = 0.0E0
C
DO 240 J = 1, L
G = 0.0E0
C .......... FORM ELEMENT OF A*U ..........
DO 180 K = 1, J
180 G = G + A(J,K) * A(I,K)
C
JP1 = J + 1
IF (L .LT. JP1) GO TO 220
C
DO 200 K = JP1, L
200 G = G + A(K,J) * A(I,K)
C .......... FORM ELEMENT OF P ..........
220 E(J) = G / H
F = F + E(J) * A(I,J)
240 CONTINUE
C
H = F / (H + H)
C .......... FORM REDUCED A ..........
DO 260 J = 1, L
F = A(I,J)
G = E(J) - H * F
E(J) = G
C
DO 260 K = 1, J
A(J,K) = A(J,K) - F * E(K) - G * A(I,K)
260 CONTINUE
C
270 DO 280 K = 1, L
280 A(I,K) = SCALE * A(I,K)
C
290 H = D(I)
D(I) = A(I,I)
A(I,I) = H
300 CONTINUE
C
RETURN
END
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