Home > wafo > trgauss > private > rqlf_asympt.m

# rqlf_asympt

## PURPOSE

Gives first two terms in an asymptotic expansion of the

## SYNOPSIS

[Cb0,Cb1,a0,a1]=rqlf_asympt(u,g,b,S22,S21)

## DESCRIPTION

```  RQLF_ASYMPT Gives first two terms in an asymptotic expansion of the
crossing rate of a quadratic+linear

CALL:  [Cb0,Cb1,a0,a1] = rqlf_asympt(u,g,b,S22,S12);

A function that gives the two first terms
in an asymptotic expansion of the rate of crossings of a
quadratic+linear form according to Breitung's method```

## CROSS-REFERENCE INFORMATION

This function calls:
 holmberg1 Computes moments for higher order reliability methods. holmberg2 Computes moments for higher order reliability methods. holmberg3 Computes moments for higher order reliability methods. holmquist1 Computes moments for higher order reliability methods. holmquist2 Computes moments for higher order reliability methods. holmquist3 Computes moments for higher order reliability methods. mindist Finds minimal distance to the origin on the surface b'*x+x'*diag(g)*x=u det Determinant. error Display message and abort function. inv Matrix inverse.
This function is called by:
 chitwo2lc_sorm SORM-approximation of the crossing intensity for the noncentral Chi^2 process

## SOURCE CODE

```001 function [Cb0,Cb1,a0,a1]=rqlf_asympt(u,g,b,S22,S21)
002 % RQLF_ASYMPT Gives first two terms in an asymptotic expansion of the
003 % crossing rate of a quadratic+linear
004 %
005 %   CALL:  [Cb0,Cb1,a0,a1] = rqlf_asympt(u,g,b,S22,S12);
006 %
007 % A function that gives the two first terms
008 % in an asymptotic expansion of the rate of crossings of a
009 % quadratic+linear form according to Breitung's method
010
011
012 g=g(:);
013 b=b(:);
014
015 if length(g)~=length(b)
016    error('The lengths of b and g should be the same.')
017 end
018 %if u<=0;
019 %   error('u should be a positive number.')
020 %end
021 n=size(S22,1);
022 xtp=mindist(g,b,u,20);
023 xJ=xtp(:);
024 b0=sqrt(sum(xJ.^2));
025 yJ=xJ/b0;
026 PJ=eye(n)-yJ*yJ';
027 dg=-b0*(b+2*(g.*xJ));
028 GJ=-2*b0^2*diag(g)/sqrt(sum(dg.^2));
029 V=S22+S21*S21;
030 sJ2=yJ'*V*yJ;
031 sJ=sqrt(sJ2);
032 a0=sJ*sqrt(1-yJ'*S21*(eye(n)+GJ)*S21*yJ/sJ2)/sqrt(det(eye(n)+ ...
033                           PJ*GJ*PJ));
034 aJ=-1/sJ*PJ*(eye(n)+GJ)*S21*yJ;
035 WJ1=inv(eye(n)+PJ*GJ*PJ+aJ*aJ');
036 WJ2=inv(eye(n)+PJ*GJ*PJ);
037 vJ1=PJ*S21*PJ*GJ*yJ;
038 vJ2=PJ*GJ*yJ;
039 vJ3=PJ*GJ*S21*yJ;
040 vJ4=PJ*GJ*PJ*V*yJ;
041 KJ1=yJ'*S21*GJ*yJ;
042 KJ2=yJ'*GJ*yJ;
043 QJ1=PJ*GJ*PJ;
044 QJ2=PJ*S21*PJ*GJ*PJ;
045 QJ3=QJ2+.5*KJ1*QJ1-vJ2*vJ3';
046 QJ4=-(yJ'*V*PJ*GJ*yJ)*PJ*GJ*PJ-2*PJ*GJ*PJ*V*yJ*yJ'*GJ*PJ+PJ*GJ* ...
047        PJ*V*PJ*GJ*PJ;
048 a11=1/2*holmquist1(WJ1,QJ4)/sJ2-1/2*holmquist1(WJ1,vJ4*vJ4')/sJ2^2+ ...
049     1/2*holmquist2(WJ1,QJ3,QJ3)/sJ2+holmquist2(WJ1,aJ*vJ4',QJ3)/ ...
050        sJ^3+1/2*holmquist2(WJ1,aJ*aJ',vJ4*vJ4')/sJ2^2+1/8* ...
051 holmquist3(WJ1,vJ2*vJ2',QJ1,QJ1)-1/2*holmquist2(WJ1,vJ2*vJ2',QJ1)-1/8*(KJ2+1)*holmquist2(WJ1,QJ1,QJ1)+...
052 1/2*holmquist2(WJ1,vJ2*vJ4',QJ1)/sJ2;
053 a12=(1/2*holmberg1(WJ2,aJ,vJ1,QJ1)+holmberg1(WJ2,aJ,vJ2,QJ2)-1/2*holmberg1(WJ2,aJ,vJ3,QJ1)+...
054     KJ1*holmberg1(WJ2,aJ,vJ2,QJ1)-holmberg1(WJ2,aJ,vJ3,vJ2*vJ2')-1/2*KJ2*holmberg1(WJ2,aJ,vJ3,QJ1))/sJ-...
055     1/2*holmberg2(WJ2,aJ,vJ2,QJ3,QJ1)/sJ-1/2*holmberg2(WJ2,aJ,aJ,vJ2*vJ2',QJ1)-...
056     1/8*(KJ2+1)*holmberg2(WJ2,aJ,aJ,QJ1,QJ1)+1/8*holmberg3(WJ2,aJ,aJ,vJ2*vJ2',QJ1,QJ1);
057 [det(WJ1) det(WJ2) sJ a11 a12];
058 a1=sJ*(sqrt(det(WJ1)*a11)+sqrt(det(WJ2))*a12);
059 Cb0=a0*exp(-b0^2/2)/pi;
060 Cb1=a1*exp(-b0^2/2)/pi/b0^2;
061
062
063
064
065
066
067
068```

Mathematical Statistics
Centre for Mathematical Sciences
Lund University with Lund Institute of Technology

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