%% get_tb
% Get scaled age and length at birth

%%
function [tau_b, lb, info] = get_tb(p, eb, lb)
  %  created at 2007/07/27 by Bas Kooijman; modified 2014/03/17, 2015/01/18
  % modified 2018/09/10 (t -> tau) Nina Marn
  
  %% Syntax
  % [tau_b, lb, info] = <../get_tb.m *get_tb*> (p, eb, lb)
  
  %% Description
  % Obtains scaled age at birth, given the scaled reserve density at birth. 
  % Divide the result by the somatic maintenance rate coefficient to arrive at age at birth. 
  %
  % Input
  %
  % * p: 1 or 3-vector with parameters g, k_J/ k_M, v_H^b
  %
  %     Last 2 values are optional in invoke call to get_lb
  %
  % * eb: optional scalar with scaled reserve density at birth (default eb = 1)
  % * lb: optional scalar with scaled length at birth (default: lb is obtained from get_lb)
  %  
  % Output
  %
  % * tau_b: scaled age at birth \tau_b = a_b k_M
  % * lb: scalar with scaled length at birth: L_b/ L_m
  % * info: indicator equals 1 if successful, 0 otherwise
  
  %% Remarks
  % See also <get_tb1.html *get_tb1*> for backward integration over maturity and
  % <get_tb_foetus.html *get_tb_foetus*> for foetal development
  
  %% Example of use
  % get_tb([.1;.5;.03])
  % See also <../mydata_ue0.m *mydata_ue0*>
   
  if ~exist('eb','var')
    eb = 1; % maximum value as juvenile
  end

  info = 1;
  
  if ~exist('lb', 'var')
    if length(p) < 3
      fprintf('not enough input parameters, see get_lb \n');
      tau_b = [];
      return;
    end
    [lb, info] = get_lb(p, eb);
  end
  if isempty(lb)
    [lb, info] = get_lb(p, eb);
  end

  % unpack p
  g = p(1);  % energy investment ratio

  xb = g/ (eb + g); % f = e_b 
  ab = 3 * g * xb^(1/ 3)/ lb; % \alpha_b
  tau_b = 3 * quad(@dget_tb, 1e-15, xb, [], [], ab, xb);
  
end

% subfunction

function f = dget_tb(x, ab, xb)
  % called by get_tb

  f = x .^ (-2/ 3) ./ (1 - x) ./ (ab - beta0(x, xb));
end