--- /dev/null
+*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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